id int64 | problem_number string | title string | statement string | background string | difficulty_level_id int64 | status string | proposed_by string | proposed_year int64 | category_id int64 | set_id int64 | view_count int64 | favorite_count int64 | created_at timestamp[s] | updated_at timestamp[s] | published bool | category dict | difficulty dict | set dict | duplicates list | status_review unknown | archive_reason string | duplicate_of list | old_problem_number string | notes string | tags list | literature_assessment string | literature_sources list | literature_checked_at timestamp[s] | solution_reference string | source_url string | source string | prize string | references list | clean_statement string | original_statement string | statement_status string | statement_verification string | research_classification string | research_summary string | research_difficulty_suggested string | historical_research_summary string | research_summary_reviewed_at timestamp[s] | source_citation string |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
801,170 | AMR-155-0009 | Open Problems in Wavelet Theory — Problem 9 | One of the most fundamental problems in wavelet theory asks for a characterization of dilations for which o.n. wavelets exist. Although this problem has been explicitly stated by Speegle [S03] and Wang [Wa], it has been studied earlier in late 1990's.
For what dilations $A \in GL_n(\mathbb{R})$ and lattices $\Gamma \s... | Source list: Bownik, Rzeszotnik - Open Problems in Wavelet Theory (2019)
Source item: Problem 9
Source URL: https://arxiv.org/abs/1905.08230
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Bownik, Rzeszotnik | 2,019 | 8 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 8,
"name": "analysis",
"display_name": "Analysis",
"description": "Limits, continuity, calculus, and function theory.",
"slug": "analysis",
"order_index": 8,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
801,171 | AMR-155-0010 | Open Problems in Wavelet Theory — Problem 10 | Does Calder\'on's formula
\begin{equation}
\sum_{j\in{\mathbb{Z}}} |\hat \psi((A^T)^j \xi)|^2=1 \qquad\text{for a.e. }\xi \in \mathbb{R}^n
\end{equation}
hold for any orthonormal (or Parseval) wavelet $\psi$ associated with $(A,\Gamma)$? | Source list: Bownik, Rzeszotnik - Open Problems in Wavelet Theory (2019)
Source item: Problem 10
Source URL: https://arxiv.org/abs/1905.08230
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Bownik, Rzeszotnik | 2,019 | 8 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 8,
"name": "analysis",
"display_name": "Analysis",
"description": "Limits, continuity, calculus, and function theory.",
"slug": "analysis",
"order_index": 8,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
801,172 | AMR-155-0011 | Open Problems in Wavelet Theory — Problem 11 | A variant of Problem 3.1 asks for a characterization of dilations for which well-localized o.n. wavelets exist. We say that a function $\psi \in L^2(\mathbb{R}^n)$ is *well-localized* if both $\psi$ and $\hat \psi$ have polynomial decay. That is, for some large $N>0$, we have
\[
\psi(x) = O(|x|^{-N}) \text{ as } |x| \t... | Source list: Bownik, Rzeszotnik - Open Problems in Wavelet Theory (2019)
Source item: Problem 11
Source URL: https://arxiv.org/abs/1905.08230
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Bownik, Rzeszotnik | 2,019 | 8 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 8,
"name": "analysis",
"display_name": "Analysis",
"description": "Limits, continuity, calculus, and function theory.",
"slug": "analysis",
"order_index": 8,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
801,173 | AMR-155-0012 | Open Problems in Wavelet Theory — Problem 12 | Schwartz class is defined as a collection of all $\infty$-regular functions on $\mathbb{R}^n$.
Do Schwartz class wavelets exist for integer expansive dilations $A$ and lattice $\Gamma ={\mathbb{Z}}^n$? | Source list: Bownik, Rzeszotnik - Open Problems in Wavelet Theory (2019)
Source item: Problem 12
Source URL: https://arxiv.org/abs/1905.08230
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Bownik, Rzeszotnik | 2,019 | 8 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 8,
"name": "analysis",
"display_name": "Analysis",
"description": "Limits, continuity, calculus, and function theory.",
"slug": "analysis",
"order_index": 8,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
801,174 | AMR-155-0013 | Open Problems in Wavelet Theory — Problem 13 | We end by stating not that serious, yet curious problem. The only known construction of wavelets in the Schwartz class is a Meyer wavelet which is a band-limited function. Hence, it is natural ask the following question.
Suppose $\psi$ is an orthonormal wavelet such that $\psi$ belongs to the Schwartz class. Is
$\hat\... | Source list: Bownik, Rzeszotnik - Open Problems in Wavelet Theory (2019)
Source item: Problem 13
Source URL: https://arxiv.org/abs/1905.08230
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Bownik, Rzeszotnik | 2,019 | 8 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 8,
"name": "analysis",
"display_name": "Analysis",
"description": "Limits, continuity, calculus, and function theory.",
"slug": "analysis",
"order_index": 8,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
801,175 | AMR-156-0001 | Stanley's Positivity Problems in Algebraic Combinatorics — Problem 1: Generalized Lower Bound Theorem and g-theorem for Gorenstein* complexes | Let $\Delta$ be a $(d-1)$-dimensional Gorenstein* simplicial complex (i.e., for every face $F\in\Delta$, including $F=\varnothing$, the reduced simplicial homology $\tilde{H}_i(\operatorname{link}(F);\mathbb{Q})$ is $\mathbb{Q}$ if $i=\dim\operatorname{link}(F)$ and $0$ otherwise), with $f$-vector $(f_0,\dots,f_{d-1})$... | Source list: Stanley - Positivity Problems and Conjectures in Algebraic Combinatorics (1999)
Source item: Problem 1
Source URL: https://math.mit.edu/~rstan/papers/problems.pdf
Accessed: 2026-10-16
Extraction: pdf+html_verified
Notes and literature:
Proved for simplicial homology spheres (and Gorenstein* complexes) by ... | 3 | solved | Stanley | 1,999 | 2 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 2,
"name": "combinatorics",
"display_name": "Combinatorics",
"description": "Counting problems, graph theory, discrete structures.",
"slug": "combinatorics",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Proved for simplicial homology spheres (and Gorenstein* complexes) by Karim Adiprasito (2018), 'Combinatorial Lefschetz theorems beyond positivity', arXiv:1812.10454, and Papadakis-Petrotou (2020), Annals of Mathemati... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
801,176 | AMR-156-0002 | Stanley's Positivity Problems in Algebraic Combinatorics — Problem 2: GLBT and nonnegativity for the toric h-vector of convex polytopes and Gorenstein* lattices | Let $\mathcal{P}$ be an arbitrary convex polytope (not necessarily rational or simplicial), or more generally a Gorenstein* poset that is a lattice with bottom element $\hat{0}$ and top element $\hat{1}$ removed. Does the toric $h$-vector (generalized $h$-vector) $h(\mathcal{P})=(h_0,h_1,\dots,h_d)$ of $\mathcal{P}$ sa... | Source list: Stanley - Positivity Problems and Conjectures in Algebraic Combinatorics (1999)
Source item: Problem 2
Source URL: https://math.mit.edu/~rstan/papers/problems.pdf
Accessed: 2026-10-16
Extraction: pdf+html_verified
Notes and literature:
Solved affirmatively for all convex polytopes (rational or nonrational... | 3 | partially_solved | Stanley | 1,999 | 2 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 2,
"name": "combinatorics",
"display_name": "Combinatorics",
"description": "Counting problems, graph theory, discrete structures.",
"slug": "combinatorics",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
801,177 | AMR-156-0003 | Stanley's Positivity Problems in Algebraic Combinatorics — Problem 4: Charney-Davis conjecture for Gorenstein* flag complexes | Let $\Delta$ be a $(2e-1)$-dimensional Gorenstein* flag simplicial complex (i.e., a Gorenstein* complex in which every minimal non-face has two vertices) with $h$-vector $(h_0,h_1,\dots,h_{2e})$. Is it true that
$$(-1)^e(h_0 - h_1 + h_2 - \cdots + h_{2e}) \ge 0?$$ | Source list: Stanley - Positivity Problems and Conjectures in Algebraic Combinatorics (1999)
Source item: Problem 4
Source URL: https://math.mit.edu/~rstan/papers/problems.pdf
Accessed: 2026-10-16
Extraction: pdf+html_verified
Notes and literature:
Conjectured by Ruth Charney and Michael Davis (Pacific J. Math. 171 (1... | 3 | open | Stanley | 1,999 | 2 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 2,
"name": "combinatorics",
"display_name": "Combinatorics",
"description": "Counting problems, graph theory, discrete structures.",
"slug": "combinatorics",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
801,178 | AMR-156-0004 | Stanley's Positivity Problems in Algebraic Combinatorics — Problem 5: Interval partitioning of simplicial complexes with acyclic low-dimensional links | Suppose that $\Delta$ is a finite abstract simplicial complex such that the link of every face $F$ of $\Delta$ of dimension at most $j$ (including $\operatorname{link}(\varnothing)=\Delta$) is acyclic over $\mathbb{Q}$. Can $\Delta$ (regarded as a poset under set inclusion) be partitioned into intervals $[F,F']$ such t... | Source list: Stanley - Positivity Problems and Conjectures in Algebraic Combinatorics (1999)
Source item: Problem 5
Source URL: https://math.mit.edu/~rstan/papers/problems.pdf
Accessed: 2026-10-16
Extraction: pdf+html_verified
Notes and literature:
Conjectured by Stanley (Discrete Math. 120 (1993), Conj. 2.4). As note... | 3 | solved | Stanley | 1,999 | 2 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 2,
"name": "combinatorics",
"display_name": "Combinatorics",
"description": "Counting problems, graph theory, discrete structures.",
"slug": "combinatorics",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Conjectured by Stanley (Discrete Math. 120 (1993), Conj. 2.4). As noted in Stanley's Positivity Problems update page, this problem has a negative answer (disproved by counterexample)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
801,179 | AMR-156-0005 | Stanley's Positivity Problems in Algebraic Combinatorics — Problem 6: Partitionability of Cohen-Macaulay simplicial complexes | Let $\Delta$ be a pure $(d-1)$-dimensional simplicial complex that is Cohen-Macaulay over $\mathbb{Q}$ (i.e., $\tilde{H}_i(\operatorname{link}(F);\mathbb{Q})=0$ for all $F\in\Delta$ and $i\ne \dim\operatorname{link}(F)$). Is every Cohen-Macaulay simplicial complex partitionable into intervals $[F,F']$ where each $F'$ i... | Source list: Stanley - Positivity Problems and Conjectures in Algebraic Combinatorics (1999)
Source item: Problem 6
Source URL: https://math.mit.edu/~rstan/papers/problems.pdf
Accessed: 2026-10-16
Extraction: pdf+html_verified
Notes and literature:
Refuted by Art M. Duval, Caroline J. Klivans, and Jeremy L. Martin, 'S... | 3 | solved | Stanley | 1,999 | 2 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 2,
"name": "combinatorics",
"display_name": "Combinatorics",
"description": "Counting problems, graph theory, discrete structures.",
"slug": "combinatorics",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Refuted by Art M. Duval, Caroline J. Klivans, and Jeremy L. Martin, 'Simplicial matrix-tree theorems' / 'Counterexamples to the partitionability conjecture', Transactions of the AMS / Notices of the AMS 64 (2017), 117... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
801,180 | AMR-156-0006 | Stanley's Positivity Problems in Algebraic Combinatorics — Problem 7: Nonnegativity of the cd-index of a Gorenstein* poset | Let $P$ be a Gorenstein* graded poset of rank $n-1$, and let $\Phi_P(c,d)$ be its $cd$-index in noncommuting variables $c$ and $d$ (where $\Psi_P(a,b)=\Phi_P(a+b,ab+ba)$ is the flag $h$-vector generating polynomial). Are all coefficients of the $cd$-index $\Phi_P(c,d)$ nonnegative? | Source list: Stanley - Positivity Problems and Conjectures in Algebraic Combinatorics (1999)
Source item: Problem 7
Source URL: https://math.mit.edu/~rstan/papers/problems.pdf
Accessed: 2026-10-16
Extraction: pdf+html_verified
Notes and literature:
Solved affirmatively by Kalle Karu, 'The cd-index of fans and posets',... | 3 | solved | Stanley | 1,999 | 2 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 2,
"name": "combinatorics",
"display_name": "Combinatorics",
"description": "Counting problems, graph theory, discrete structures.",
"slug": "combinatorics",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved affirmatively by Kalle Karu, 'The cd-index of fans and posets', Compos. Math. 142 (2006), 701-718 (arXiv:math/0410513)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
801,181 | AMR-156-0007 | Stanley's Positivity Problems in Algebraic Combinatorics — Problem 8: Adin's cubical h-vector nonnegativity and GLBT for cubical complexes | Let $L$ be a pure cubical complex of rank $d$ (dimension $d-1$), and let $h(L)=(h_0(L),h_1(L),\dots,h_d(L))$ be Adin's cubical $h$-vector defined by $\sum_{i=0}^d h_i(L)x^i = \frac{1}{1+x}\left(2^{d-1} + x\sum_s h(\Delta_s,x) + (-2)^{d-1}\widetilde{\chi}(L)x^{d+1}\right)$, where $\Delta_s=\operatorname{link}(s)$ for ea... | Source list: Stanley - Positivity Problems and Conjectures in Algebraic Combinatorics (1999)
Source item: Problem 8
Source URL: https://math.mit.edu/~rstan/papers/problems.pdf
Accessed: 2026-10-16
Extraction: pdf+html_verified
Notes and literature:
Raised by Ron M. Adin (Discrete Math. 157 (1996), 3-14), who proved pa... | 3 | open | Stanley | 1,999 | 2 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 2,
"name": "combinatorics",
"display_name": "Combinatorics",
"description": "Counting problems, graph theory, discrete structures.",
"slug": "combinatorics",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
801,182 | AMR-156-0008 | Stanley's Positivity Problems in Algebraic Combinatorics — Problem 9: Combinatorial interpretation of plethysm coefficients | Let $s_\lambda$ denote the Schur function indexed by a partition $\lambda$, and let $s_m[s_n]$ (or more generally $s_\lambda[s_\mu]$) denote the plethysm of Schur functions. Find a positive combinatorial interpretation of the plethysm coefficients $\langle s_m[s_n], s_\lambda\rangle$, thereby combinatorially reproving ... | Source list: Stanley - Positivity Problems and Conjectures in Algebraic Combinatorics (1999)
Source item: Problem 9
Source URL: https://math.mit.edu/~rstan/papers/problems.pdf
Accessed: 2026-10-16
Extraction: pdf+html_verified
Notes and literature:
One of the central open problems in algebraic combinatorics (Stanley's... | 3 | open | Stanley | 1,999 | 2 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 2,
"name": "combinatorics",
"display_name": "Combinatorics",
"description": "Counting problems, graph theory, discrete structures.",
"slug": "combinatorics",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
801,183 | AMR-156-0009 | Stanley's Positivity Problems in Algebraic Combinatorics — Problem 10: Combinatorial interpretation of Kronecker product coefficients of symmetric group characters | Let $\chi^\lambda$ and $\chi^\mu$ be irreducible characters of the symmetric group $S_n$ indexed by partitions $\lambda,\mu\vdash n$, and let $g_{\lambda\mu\nu} = \langle \chi^\lambda \chi^\mu, \chi^\nu\rangle$ be the Kronecker product coefficients, equivalently defined by $\prod_{i,j,k}(1-x_i y_j z_k)^{-1} = \sum_{\la... | Source list: Stanley - Positivity Problems and Conjectures in Algebraic Combinatorics (1999)
Source item: Problem 10
Source URL: https://math.mit.edu/~rstan/papers/problems.pdf
Accessed: 2026-10-16
Extraction: pdf+html_verified
Notes and literature:
Stanley's Problem 10; central to algebraic combinatorics and Geometri... | 3 | open | Stanley | 1,999 | 2 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 2,
"name": "combinatorics",
"display_name": "Combinatorics",
"description": "Counting problems, graph theory, discrete structures.",
"slug": "combinatorics",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
801,184 | AMR-156-0010 | Stanley's Positivity Problems in Algebraic Combinatorics — Problem 11: Combinatorial interpretation of Schubert polynomial structure constants | Let $\mathfrak{S}_w$ denote the Schubert polynomial indexed by a permutation $w\in S_\infty$. In the polynomial ring $\mathbb{Z}[x_1,x_2,\dots]$, the product of two Schubert polynomials expands as $\mathfrak{S}_u \mathfrak{S}_v = \sum_w c_{uv}^w \mathfrak{S}_w$, where the structure constants $c_{uv}^w$ are known to be ... | Source list: Stanley - Positivity Problems and Conjectures in Algebraic Combinatorics (1999)
Source item: Problem 11
Source URL: https://math.mit.edu/~rstan/papers/problems.pdf
Accessed: 2026-10-16
Extraction: pdf+html_verified
Notes and literature:
Stanley's Problem 11; known in special cases (Grassmannian, 2-step fl... | 3 | open | Stanley | 1,999 | 2 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 2,
"name": "combinatorics",
"display_name": "Combinatorics",
"description": "Counting problems, graph theory, discrete structures.",
"slug": "combinatorics",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
801,185 | AMR-156-0011 | Stanley's Positivity Problems in Algebraic Combinatorics — Problem 12: Combinatorial interpretation of the row sums of the character table of S_n | Let $S_n$ act on itself by conjugation, with character $\psi_{S_n}$. The multiplicity of an irreducible character $\chi^\lambda$ of $S_n$ in $\psi_{S_n}$ is equal to the row sum $\sum_K \chi^\lambda(K)$ over all conjugacy classes $K$ of $S_n$, which is therefore a nonnegative integer. Give a combinatorial interpretatio... | Source list: Stanley - Positivity Problems and Conjectures in Algebraic Combinatorics (1999)
Source item: Problem 12
Source URL: https://math.mit.edu/~rstan/papers/problems.pdf
Accessed: 2026-10-16
Extraction: pdf+html_verified
Notes and literature:
Stanley's Problem 12 (see also Enumerative Combinatorics, Vol. 2, Exe... | 3 | open | Stanley | 1,999 | 2 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 2,
"name": "combinatorics",
"display_name": "Combinatorics",
"description": "Counting problems, graph theory, discrete structures.",
"slug": "combinatorics",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
801,186 | AMR-156-0012 | Stanley's Positivity Problems in Algebraic Combinatorics — Problem 13: Positivity of the (q,t)-Kostka polynomials | Let $K_{\lambda\mu}(q,t)$ denote the $(q,t)$-Kostka polynomials defined via the integral form Macdonald symmetric functions $J_\mu(x;q,t) = \sum_\lambda K_{\lambda\mu}(q,t) s_\lambda[X(1-t)]$. Show that $K_{\lambda\mu}(q,t) \in \mathbb{N}[q,t]$, i.e., that $K_{\lambda\mu}(q,t)$ has nonnegative integer coefficients. | Source list: Stanley - Positivity Problems and Conjectures in Algebraic Combinatorics (1999)
Source item: Problem 13
Source URL: https://math.mit.edu/~rstan/papers/problems.pdf
Accessed: 2026-10-16
Extraction: pdf+html_verified
Notes and literature:
Solved by Mark Haiman, 'Hilbert schemes, polygraphs and the Macdonald... | 3 | solved | Stanley | 1,999 | 2 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 2,
"name": "combinatorics",
"display_name": "Combinatorics",
"description": "Counting problems, graph theory, discrete structures.",
"slug": "combinatorics",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved by Mark Haiman, 'Hilbert schemes, polygraphs and the Macdonald positivity conjecture', Journal of the American Mathematical Society 14 (2001), 941-1006 (arXiv:math.AG/0010246)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
801,187 | AMR-156-0013 | Stanley's Positivity Problems in Algebraic Combinatorics — Problem 14: Combinatorial symmetry and Schur-positivity of LLT ribbon polynomials | For a skew shape $\lambda/\mu$ tileable by $k$-ribbons, let $H^{(k)}_\lambda(x;q) = \sum_T q^{s(T)} x^T$ be the Lascoux-Leclerc-Thibon (LLT) $k$-ribbon generating polynomial summed over $k$-ribbon tableaux $T$ of shape $\lambda$ with spin $s(T)$.
(a) Give a combinatorial proof that the polynomials $H^{(k)}_\lambda(x;q... | Source list: Stanley - Positivity Problems and Conjectures in Algebraic Combinatorics (1999)
Source item: Problem 14
Source URL: https://math.mit.edu/~rstan/papers/problems.pdf
Accessed: 2026-10-16
Extraction: pdf+html_verified
Notes and literature:
Part (b) (Schur-positivity of LLT polynomials) was proved using Kazhd... | 3 | partially_solved | Stanley | 1,999 | 2 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 2,
"name": "combinatorics",
"display_name": "Combinatorics",
"description": "Counting problems, graph theory, discrete structures.",
"slug": "combinatorics",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
801,188 | AMR-156-0014 | Stanley's Positivity Problems in Algebraic Combinatorics — Problem 15: Kazhdan-Lusztig polynomial nonnegativity for arbitrary Coxeter groups | Let $W$ be an arbitrary Coxeter group and let $u \le v$ be elements in the Bruhat order of $W$. Are the coefficients of the Kazhdan-Lusztig polynomial $P_{u,v}(q)$ nonnegative integers? | Source list: Stanley - Positivity Problems and Conjectures in Algebraic Combinatorics (1999)
Source item: Problem 15
Source URL: https://math.mit.edu/~rstan/papers/problems.pdf
Accessed: 2026-10-16
Extraction: pdf+html_verified
Notes and literature:
Solved affirmatively in full generality by Ben Elias and Geordie Will... | 3 | solved | Stanley | 1,999 | 2 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 2,
"name": "combinatorics",
"display_name": "Combinatorics",
"description": "Counting problems, graph theory, discrete structures.",
"slug": "combinatorics",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved affirmatively in full generality by Ben Elias and Geordie Williamson, 'The Hodge theory of Soergel bimodules', Annals of Mathematics 180 (2014), 1089-1136 (arXiv:1212.0791)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
801,189 | AMR-156-0015 | Stanley's Positivity Problems in Algebraic Combinatorics — Problem 16: Combinatorial interpretation of Kazhdan-Lusztig polynomial coefficients for S_n | Give a direct positive combinatorial interpretation of the coefficients of the Kazhdan-Lusztig polynomial $P_{u,v}(q)$ when $W$ is a finite or affine Weyl group, especially for the symmetric group $W = S_n$, thereby combinatorially reproving that they are nonnegative. | Source list: Stanley - Positivity Problems and Conjectures in Algebraic Combinatorics (1999)
Source item: Problem 16
Source URL: https://math.mit.edu/~rstan/papers/problems.pdf
Accessed: 2026-10-16
Extraction: pdf+html_verified
Notes and literature:
Stanley's Problem 16. Known for special classes of permutations (such... | 3 | open | Stanley | 1,999 | 2 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 2,
"name": "combinatorics",
"display_name": "Combinatorics",
"description": "Counting problems, graph theory, discrete structures.",
"slug": "combinatorics",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
801,190 | AMR-156-0016 | Stanley's Positivity Problems in Algebraic Combinatorics — Problem 17: Stembridge's monomial immanant positivity conjecture for totally nonnegative matrices | For a partition $\lambda \vdash n$, let $\phi^\lambda$ denote the unique class function on $S_n$ whose Frobenius characteristic $\operatorname{ch}\phi^\lambda = \frac{1}{n!}\sum_{w\in S_n}\phi^\lambda(w)p_{\rho(w)}$ is the monomial symmetric function $m_\lambda$, and define the immanant $\operatorname{Imm}_{\phi^\lambd... | Source list: Stanley - Positivity Problems and Conjectures in Algebraic Combinatorics (1999)
Source item: Problem 17
Source URL: https://math.mit.edu/~rstan/papers/problems.pdf
Accessed: 2026-10-16
Extraction: pdf+html_verified
Notes and literature:
Conjectured by John R. Stembridge (Bull. London Math. Soc. 23 (1991),... | 3 | open | Stanley | 1,999 | 2 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 2,
"name": "combinatorics",
"display_name": "Combinatorics",
"description": "Counting problems, graph theory, discrete structures.",
"slug": "combinatorics",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
801,191 | AMR-156-0017 | Stanley's Positivity Problems in Algebraic Combinatorics — Problem 17': h-positivity of the permanent-like generating symmetric function of a totally nonnegative matrix | Let $A=(a_{ij})$ be a totally nonnegative $n\times n$ real matrix, and define the symmetric function
$$F_A = \sum_{w\in S_n} a_{1,w(1)}a_{2,w(2)}\cdots a_{n,w(n)} p_{\rho(w)},$$
where $p_{\rho(w)}$ is the power sum symmetric function indexed by the cycle type of $w$. Is $F_A$ $h$-positive (i.e., does its expansion in t... | Source list: Stanley - Positivity Problems and Conjectures in Algebraic Combinatorics (1999)
Source item: Problem 17'
Source URL: https://math.mit.edu/~rstan/papers/problems.pdf
Accessed: 2026-10-16
Extraction: pdf+html_verified
Notes and literature:
Equivalent formulation of Problem 17 (Stanley, Enumerative Combinato... | 3 | open | Stanley | 1,999 | 2 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 2,
"name": "combinatorics",
"display_name": "Combinatorics",
"description": "Counting problems, graph theory, discrete structures.",
"slug": "combinatorics",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
801,192 | AMR-156-0018 | Stanley's Positivity Problems in Algebraic Combinatorics — Problem 18: Stembridge's Schur-positivity conjecture for monomial immanants of Jacobi-Trudi matrices | Let $\mu$ and $\nu$ be partitions with $\nu \subseteq \mu$ and $\ell(\mu) \le n$, and let $H_{\mu/\nu} = [h_{\mu_i - \nu_j + j - i}]_{i,j=1}^n$ be the Jacobi-Trudi matrix. For any partition $\lambda \vdash n$, is the monomial immanant $\operatorname{Imm}_{\phi^\lambda}(H_{\mu/\nu})$ Schur-positive ($s$-positive)? | Source list: Stanley - Positivity Problems and Conjectures in Algebraic Combinatorics (1999)
Source item: Problem 18
Source URL: https://math.mit.edu/~rstan/papers/problems.pdf
Accessed: 2026-10-16
Extraction: pdf+html_verified
Notes and literature:
Conjectured by John R. Stembridge (Canad. J. Math. 44 (1992), 1079-10... | 3 | open | Stanley | 1,999 | 2 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 2,
"name": "combinatorics",
"display_name": "Combinatorics",
"description": "Counting problems, graph theory, discrete structures.",
"slug": "combinatorics",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
801,193 | AMR-156-0019 | Stanley's Positivity Problems in Algebraic Combinatorics — Problem 19: Haiman's unimodality and positivity conjecture for Hecke algebra monomial characters on Kazhdan-Lusztig basis elements | Let $H_n(q)$ be the Iwahori-Hecke algebra of type $A_{n-1}$, let $\phi^\lambda$ be the monomial virtual character of $H_n(q)$ corresponding to $\lambda \vdash n$, and let $C'_w$ be the Kazhdan-Lusztig basis element indexed by $w\in S_n$. Is it true that $\phi^\lambda(q^{\ell(w)/2} C'_w)$ is a polynomial in $q$ with non... | Source list: Stanley - Positivity Problems and Conjectures in Algebraic Combinatorics (1999)
Source item: Problem 19
Source URL: https://math.mit.edu/~rstan/papers/problems.pdf
Accessed: 2026-10-16
Extraction: pdf+html_verified
Notes and literature:
Conjectured by Mark Haiman (J. Amer. Math. Soc. 6 (1993), 569-595); S... | 3 | open | Stanley | 1,999 | 2 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 2,
"name": "combinatorics",
"display_name": "Combinatorics",
"description": "Counting problems, graph theory, discrete structures.",
"slug": "combinatorics",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
801,194 | AMR-156-0020 | Stanley's Positivity Problems in Algebraic Combinatorics — Problem 20: Neggers-Stanley Poset Conjecture on real zeros of the P-Eulerian polynomial W(P,x) | Let $P$ be a finite poset on $n$ elements, let $\mathcal{L}(P)$ denote the set of linear extensions of $P$ (for a natural labeling, or via $P$-partitions in general), and let $W(P,x) = \sum_{w\in\mathcal{L}(P)} x^{d(w)}$ be the order polynomial generating the descent statistic $d(w)$. Is every zero of $W(P,x)$ real? | Source list: Stanley - Positivity Problems and Conjectures in Algebraic Combinatorics (1999)
Source item: Problem 20
Source URL: https://math.mit.edu/~rstan/papers/problems.pdf
Accessed: 2026-10-16
Extraction: pdf+html_verified
Notes and literature:
Disproved by Petter Branden (Electronic Research Announcements of the... | 3 | solved | Stanley | 1,999 | 2 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 2,
"name": "combinatorics",
"display_name": "Combinatorics",
"description": "Counting problems, graph theory, discrete structures.",
"slug": "combinatorics",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Disproved by Petter Branden (Electronic Research Announcements of the AMS 10 (2004), 125-128) for general labeled posets, and subsequently disproved for naturally labeled posets by John R. Stembridge, 'Counterexamples... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
801,195 | AMR-156-0021 | Stanley's Positivity Problems in Algebraic Combinatorics — Problem 20': Real-rootedness of the chain polynomial of a finite distributive lattice | For a finite poset $P$, let $c_k(P)$ denote the number of $k$-element chains in $P$ (with $c_0=1$), and define the chain polynomial $C(P,t) = \sum_{k\ge 0} c_k(P) t^k$. When $L$ is a finite distributive lattice, is every zero of $C(L,t)$ real? | Source list: Stanley - Positivity Problems and Conjectures in Algebraic Combinatorics (1999)
Source item: Problem 20'
Source URL: https://math.mit.edu/~rstan/papers/problems.pdf
Accessed: 2026-10-16
Extraction: pdf+html_verified
Notes and literature:
Equivalent to the naturally labeled case of Problem 20 (Brenti 1989)... | 3 | solved | Stanley | 1,999 | 2 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 2,
"name": "combinatorics",
"display_name": "Combinatorics",
"description": "Counting problems, graph theory, discrete structures.",
"slug": "combinatorics",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Equivalent to the naturally labeled case of Problem 20 (Brenti 1989); disproved by John R. Stembridge, Transactions of the AMS 359 (2007), 1115-1128."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
801,196 | AMR-156-0022 | Stanley's Positivity Problems in Algebraic Combinatorics — Problem 21: Stanley-Stembridge e-positivity conjecture for chromatic symmetric functions of (3+1)-free posets | Let $P$ be a finite $(3+1)$-free poset (i.e., $P$ contains no induced subposet isomorphic to the disjoint union of a $3$-element chain and a $1$-element chain), and let $G = \operatorname{inc}(P)$ be its incomparability graph. Is the chromatic symmetric function $X_G(x) = \sum_\kappa x_1^{a_1(\kappa)} x_2^{a_2(\kappa)}... | Source list: Stanley - Positivity Problems and Conjectures in Algebraic Combinatorics (1999)
Source item: Problem 21
Source URL: https://math.mit.edu/~rstan/papers/problems.pdf
Accessed: 2026-10-16
Extraction: pdf+html_verified
Notes and literature:
Conjectured by Richard P. Stanley and John R. Stembridge (J. Combin. ... | 3 | open | Stanley | 1,999 | 2 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 2,
"name": "combinatorics",
"display_name": "Combinatorics",
"description": "Counting problems, graph theory, discrete structures.",
"slug": "combinatorics",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
801,197 | AMR-156-0023 | Stanley's Positivity Problems in Algebraic Combinatorics — Problem 22: Gasharov's Schur-positivity conjecture for chromatic symmetric functions of claw-free graphs | Let $G$ be a finite claw-free graph (i.e., $G$ has no induced subgraph isomorphic to $K_{1,3}$). Is the chromatic symmetric function $X_G(x)$ of $G$ Schur-positive ($s$-positive)? | Source list: Stanley - Positivity Problems and Conjectures in Algebraic Combinatorics (1999)
Source item: Problem 22
Source URL: https://math.mit.edu/~rstan/papers/problems.pdf
Accessed: 2026-10-16
Extraction: pdf+html_verified
Notes and literature:
Conjectured by Vesselin Gasharov (reported in Stanley, Discrete Math.... | 3 | open | Stanley | 1,999 | 2 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 2,
"name": "combinatorics",
"display_name": "Combinatorics",
"description": "Counting problems, graph theory, discrete structures.",
"slug": "combinatorics",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
801,198 | AMR-156-0024 | Stanley's Positivity Problems in Algebraic Combinatorics — Problem 23: Real-rootedness of the independence (stable set) polynomial of a claw-free graph | For a finite graph $G$, let $c_k$ be the number of stable (independent) $k$-element subsets of vertices of $G$, and define the stable set (independence) polynomial $D(G,t) = \sum_{k\ge 0} c_k t^k$. If $G$ is a claw-free graph, does $D(G,t)$ have only real zeros? | Source list: Stanley - Positivity Problems and Conjectures in Algebraic Combinatorics (1999)
Source item: Problem 23
Source URL: https://math.mit.edu/~rstan/papers/problems.pdf
Accessed: 2026-10-16
Extraction: pdf+html_verified
Notes and literature:
Solved affirmatively by Maria Chudnovsky and Paul Seymour, 'The roots... | 3 | solved | Stanley | 1,999 | 2 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 2,
"name": "combinatorics",
"display_name": "Combinatorics",
"description": "Counting problems, graph theory, discrete structures.",
"slug": "combinatorics",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved affirmatively by Maria Chudnovsky and Paul Seymour, 'The roots of the independence polynomial of a clawfree graph', Journal of Combinatorial Theory, Series B 97 (2007), 350-357."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
801,199 | AMR-156-0025 | Stanley's Positivity Problems in Algebraic Combinatorics — Problem 24: Haglund-Ono-Wagner real-rootedness conjecture for per(A + xJ) | Let $A$ be an $n\times n$ real matrix for which every column is weakly increasing (and nonnegative), and let $J$ denote the $n\times n$ matrix of all $1$'s. Does the rook/permanent polynomial $\operatorname{per}(A + xJ)$ have only real zeros? | Source list: Stanley - Positivity Problems and Conjectures in Algebraic Combinatorics (1999)
Source item: Problem 24
Source URL: https://math.mit.edu/~rstan/papers/problems.pdf
Accessed: 2026-10-16
Extraction: pdf+html_verified
Notes and literature:
Solved affirmatively by Petter Branden, James Haglund, Mirko Visontai... | 3 | solved | Stanley | 1,999 | 2 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 2,
"name": "combinatorics",
"display_name": "Combinatorics",
"description": "Counting problems, graph theory, discrete structures.",
"slug": "combinatorics",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved affirmatively by Petter Branden, James Haglund, Mirko Visontai, and David G. Wagner, 'Proof of the monotone column permanent conjecture', in Notions of Positivity and the Geometry of Polynomials (Birkhauser, 20... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
801,200 | AMR-156-0026 | Stanley's Positivity Problems in Algebraic Combinatorics — Problem 25: Unimodality and log-concavity of matroid and geometric lattice sequences (Read-Hoggar-Welsh-Mason conjectures) | Are the following sequences unimodal or log-concave for every finite graph or matroid?
(a) The absolute values of the coefficients of the chromatic polynomial of a graph, or more generally, the characteristic polynomial of a matroid.
(b) The number of $i$-edge spanning forests of a graph, or more generally, the numbe... | Source list: Stanley - Positivity Problems and Conjectures in Algebraic Combinatorics (1999)
Source item: Problem 25
Source URL: https://math.mit.edu/~rstan/papers/problems.pdf
Accessed: 2026-10-16
Extraction: pdf+html_verified
Notes and literature:
All three parts have been proved in landmark works using Hodge theory... | 3 | solved | Stanley | 1,999 | 2 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 2,
"name": "combinatorics",
"display_name": "Combinatorics",
"description": "Counting problems, graph theory, discrete structures.",
"slug": "combinatorics",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "All three parts have been proved in landmark works using Hodge theory of matroids: (a) proved by June Huh (2012) for realizable matroids and Karim Adiprasito, June Huh, and Eric Katz, 'Hodge theory for combinatorial g... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
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