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Symplectic $Q$-functions are a symplectic analogue of Schur $Q$-functions and defined as the $t=-1$ specialization of Hall--Littlewood functions associated with the root system of type $C$. In this paper we prove that symplectic $Q$-functions share many of the properties of Schur $Q$-functions, such as a tableau descr... |
We extend the convergence law for sparse random graphs proven by Lynch to arbitrary relational languages. We consider a finite relational vocabulary $\sigma$ and a first order theory $T$ for $\sigma$ composed of symmetry and anti-reflexivity axioms. |
We study (a:a) Maker-Breaker games played on the edge set of the complete graph on n vertices. In the following four games - perfect matching game, Hamilton cycle game, star factor game and path factor game, our goal is to determine the least number of moves which Maker needs in order to win these games. |
We consider multifold convolutions of a combinatorial sequence $(a_n)_{n=0}^{\infty}$: namely, for each $k \in \N$ the $k$-fold convolution is $\mathcal{M}^{(k)}_n(\boldsymbol{a}) = \sum_{j_1+\dots+j_k=n} a_{j_1} \cdots a_{j_k}$. Let $C_n$ be the Catalan numbers, and let $B_n$ be the central binomial coefficients. |
Prüfer codes provide an encoding scheme for representing a vertex-labeled tree on $n$ vertices with a string of length $n-2$. Indeed, two labeled trees are isomorphic if and only if their Prüfer codes are identical, and this supplies a proof of Cayley's Theorem. |
The Binary Two-Up Sequence is the lexicographically earliest sequence of distinct nonnegative integers with the property that the binary expansion of the n-th term has no 1-bits in common with any of the previous floor(n/2) terms. We show that the sequence can be decomposed into ``atoms'', which are sequences ... |
Let $\mathcal{A}(H)$ be the adjacency tensor of $r$-uniform hypergraph $H$. If $H$ is connected, the unique positive eigenvector $x=(x_1,x_2,\ldots,x_n)^{\mathrm{T}}$ with $||x||_r=1$ corresponding to spectral radius $\rho(H)$ is called the principal eigenvector of $H$. |
Recently, settling a question of Erdős, Balogh and Petříčková showed that there are at most $2^{n^2/8+o(n^2)}$ $n$-vertex maximal triangle-free graphs, matching the previously known lower bound. Here we characterize the typical structure of maximal triangle-free graphs. |
The set of hook lengths of an integer partition $\lambda$ is the complement of some numerical semigroup $S$. There has been recent interest in studying the number of partitions with a given set of hook lengths. |
We prove that there exists a constant $c>0$ such that for all integers $2\leq t\leq cn$, if $\calA$ is a collection of spanning trees in $K_n$ such that any two intersect at at least $t$ edges, then $|\calA|\leq 2^tn^{n-t-2}$. This bound is tight; the equality is achieved when $\calA$ is a collection of spanning tr... |
There exist few examples of negative Latin square type partial difference sets (NLST PDSs) in nonabelian groups. We present a list of 176 inequivalent NLST PDSs in 48 nonisomorphic, nonabelian groups of order 64. |
The aim of this paper is to give a corrected bijective proof of Vershik's relations for the Kostka numbers. Our proof uses insertion and reverse insertion algorithms, as in the combinatorial proof of the Pieri rule. |
A graph $G$ is geodetic if between any two vertices there exists a unique shortest path. In 1962 Ore raised the challenge to characterize geodetic graphs, but despite many attempts, such characterization still seems well beyond reach. |
What distribution of graphical degree sequence is invariant under ``scaling''? Are these graphs always power-law graphs? |
We consider the tiling of an $n$-board (a $1\times n$ array of square cells of unit width) with half-squares ($\frac12\times1$ tiles) and $(\frac12,\frac12)$-fence tiles. A $(\frac12,\frac12)$-fence tile is composed of two half-squares separated by a gap of width $\frac12$. |
We prove lower bounds for the fraction of edges of an $r$-graph which can be covered by the union of $k$ 1-factors. The special case $r=3$ yields some known results for cubic graphs. |
Given a graph $G$, a coloring $c:V(G)\longrightarrow \{1,\ldots,k\}$ such that $c (u)=c |
A cyclic base ordering of a connected graph $G$, is a cyclic ordering of $E(G)$ such that every cyclically consecutive $|V(G)|-1$ edges form a spanning tree. In this project, we study cyclic base ordering of various families of graphs, including square of cycles, wheel graphs, generalized wheel graphs and broken wheel... |
A set $A\subseteq\mathbb N$ is called $complete$ if every sufficiently large integer can be written as the sum of distinct elements of $A$. In this paper we present a new method for proving the completeness of a set, improving results of Cassels ('60), Zannier ('92), Burr, Erdős, Graham, and Li ('96), and ... |
We introduce and investigate $d$-convex union representable complexes: the complexes that arise as the nerve of a finite collection of convex open sets in $\mathbb R^d$ whose union is also convex. Chen, Frick, and Shiu recently proved that such complexes are collapsible and asked if all collapsible complexes are conve... |
A reconstruction problem is formulated for multisets over commutative groupoids. The cards of a multiset are obtained by replacing a pair of its elements by their sum. |
Given a graph $G$ consider a procedure of building a dominating set $D$ in $G$ by adding vertices to $D$ one at a time in such a way that whenever vertex $x$ is added to $D$ there exists a vertex $y\in N_G[x]$ that becomes dominated only after $x$ is added to $D$. The maximum cardinality of a set $D$ obtained in the d... |
Three--dimensional colored triangulations are gluings of tetrahedra whose faces carry the colors 0, 1, 2, 3 and in which the attaching maps between tetrahedra are defined using the colors. This framework makes it possible to generalize the notion of two--dimensional $2p$--angulations to three dimensions in a way which ... |
Let $k\ge 1$ be an odd integer, $t=\lfloor {{k+2}\over 4}\rfloor$, and $q$ be a prime power. We construct a bipartite, $q$-regular, edge-transitive graph $C\! |
In this paper, we show that every D3-directing CNFA can be mapped uniquely to a DFA with the same synchronizing word length. This implies that Černý's conjecture generalizes to CNFAs and that the general upper bound for the length of a shortest D3-directing word is equal to the Pin-Frankl bound for DFAs. |
A digraph $D=(V,A)$ of order $n\geq 3$ is pancyclic, whenever $D$ contains a directed cycle of length $k$ for each $k\in\{3,...,n\}$; and D is vertex-pancyclic iff, for each vertex $v\in V$ and each $k\in \{3,...,n\}$, $D$ contains a directed cycle of length $k$ passing through $v$. <br>Let $D_1$, $D_2$,..., $D_k$ be ... |
We determine the colored patterns that appear in any $2$-edge coloring of $K_{n,n}$, with $n$ large enough and with sufficient edges in each color. We prove the existence of a positive integer $z_2$ such that any $2$-edge coloring of $K_{n,n}$ with at least $z_2$ edges in each color contains at least one of these patt... |
We refine a property of $2$-connected graphs described in the classical paper of Dirac from 1952 and use the refined property to somewhat shorten Dirac's proof of the fact that each $2$-connected $n$-vertex graph with minimum degree at least $k$ has a cycle of length at least $\min\{n,2k\}$. |
Let $S_k(n)$ be the maximum number of orientations of an $n$-vertex graph $G$ in which no copy of $K_k$ is strongly connected. For all integers $n$, $k\geq 4$ where $n\geq 5$ or $k\geq 5$, we prove that $S_k(n) = 2^{t_{k-1}(n)}$, where $t_{k-1}(n)$ is the number of edges of the $n$-vertex $(k-1)$-partite Turán graph $... |
The ring of symmetric functions has a basis of dual Grothendieck polynomials that are inhomogeneous $K$-theoretic deformations of Schur polynomials. We prove that dual Grothendieck polynomials determine column distributions for a directed last-passage percolation model. |
In this paper we study the operation of cutting off edges of a simple $3$-polytope $P$ along the graph $\Gamma$. We give the criterion when the resulting polytope is simple and when it is flag. |
We generalize the definition and enumeration of spanning trees from the setting of graphs to that of arbitrary-dimensional simplicial complexes $\Delta$, extending an idea due to G. Kalai. We prove a simplicial version of the Matrix-Tree Theorem that counts simplicial spanning trees, weighted by the squares of the ord... |
In this paper we prowide lower bounds on the complexity of the DNF exception problem for short exception lists and hypercube covering problem. The method proposed is based on the relaxation of the initial problem to a certain linear programming problem. |
If an $n$-uniform hypergraph can be 2-colored, then it is said to have property B. Erdős (1963) was the first to give lower and upper bounds for the minimal size $m(n)$ of an $n$-uniform hypergraph without property B. His asymptotic upper bound $O(n^22^n)$ still is the best we know, his lower bound $2^{n-1}$ has seen a... |
We introduce the hydra continued fractions, as a generalization of the Rogers-Ramanujan continued fractions, and give a combinatorial interpretation in terms of shift-plethystic trees. We then show it is possible to express them as a quotient of m-distinct partition generating functions, and in its dual form as a quot... |
We consider the problem of constructing Latin cubes subject to the condition that some symbols may not appear in certain cells. We prove that there is a constant $\gamma > 0$ such that if $n=2t$ and $A$ is a $3$-dimensional $n\times n\times n$ array where every cell contains at most $\gamma n$ symbols, and every sy... |
An s-cap n-flat, or an n-dimensional cap of size s, is a pair (S,F) where F is an n-dimensional affine space over Z/3Z and the size-s subset S of F contains no triple of collinear points. The cap set problem in dimension n asks for the largest s for which an s-cap n-flat exists. |
We investigate a family of polytopes introduced by E.M.\ Feichtner, A. \ Postnikov and B. |
We show that certain Riordan arrays have generating functions that can be expressed as continued fractions of Jacobi and Thron type. We investigate the inverses of such arrays, which in certain circumstances can also have generating functions representable as continued fractions. |
Assume G is a finite group, such that |G|= 6pq or 7pq, where p and q are distinct prime numbers, and let S be a generating set of G. We prove there is a Hamiltonian cycle in the corresponding Cayley graph Cay(G;S). |
We show how multiplicatively syndetic sets can be used in the study of partition regularity of dilation invariant systems of polynomial equations. In particular, we prove that a dilation invariant system of polynomial equations is partition regular if and only if it has a solution inside every multiplicatively syndeti... |
We investigate bounds on the dichromatic number of digraphs which avoid a fixed digraph as a topological minor. For a digraph $F$, denote by $\text{mader}_{\vec{\chi}}(F)$ the smallest integer $k$ such that every $k$-dichromatic digraph contains a subdivision of $F$. |
Let $\bm{i}=1+q+...+q^{i-1}$. For certain sequences $(r_1,...,r_l)$ of positive integers, we show that in the Hecke algebra $\mathscr{H}_n(q)$ of the symmetric group $\mathfrak{S}_n$, the product $(1+\bm{r_1}T_{r_1})... (1+\bm{r_l}T_{r_l})$ has a simple explicit expansion in terms of the standard basis $\{T_w\}$. |
The Oberwolfach Problem $OP(F)$ -- posed by Gerhard Ringel in 1967 -- is a paradigmatic Combinatorial Design problem asking whether the complete graph $K_v$ decomposes into edge-disjoint copies of a $2$-regular graph $F$ of order $v$. In this paper, we provide all the necessary equipment to generate solutions to $OP(F... |
In this article, we define a family of regular bipartite graphs and show that the homotopy type of the independence complexes of this family is the wedge sum of spheres of certain dimensions. |
Let $\mathcal G$ be a hypergraph whose edges are colored. An {\it $(\alpha,n)$-detachment} of $\mathcal G$ is a hypergraph obtained by splitting a vertex $\alpha$ into $n$ vertices, say $\alpha_1,\dots,\alpha_n$, and sharing the incident hinges and edges among the subvertices. |
The Catalan-Larcombe-French sequence $\{P_n\}_{n\geq 0}$ arises in a series expansion of the complete elliptic integral of the first kind. It has been proved that the sequence is log-balanced. |
In 1996, Youngs proved that any quadrangulation of the real projective plane is not 3-chromatic. This result has been extended in various directions over the years, including to other non-orientable closed surfaces, higher-dimensional analogues of quadrangulations and circular colorings. |
We generalise the results by Bigorajska and Kotlarski about partitioning $\alpha$-large sets, by extending the domain up to ordinals below $\varepsilon_{\omega}$. These results will be very useful to give a miniaturisation of the infinite Ramsey Theorem. |
{Let ${\Cal X}$ be a self-dual P-polynomial association scheme. Then there are at most 12 diagonal matrices $T$ such that $(PT)^3=I$. |
For two graphs, $G$, and $H$, an edge-coloring of a complete graph is $(G,H)$-good if there is no monochromatic subgraph isomorphic to $G$ and no rainbow subgraph isomorphic to $H$ in this coloring. The set of number of colors used by some $(G,H)$-colorings of $K_n$ is called a mixed-Ramsey spectrum. |
The dominating number $\gamma(G)$ of a graph $G$ is the minimum size of a vertex set whose closed neighborhood covers all the vertices of the graph. The packing number $\rho(G)$ of $G$ is the maximum size of a vertex set whose closed neighborhoods are pairwise disjoint. |
In previous work, we associated to any finite simple graph a particular set of derangements of its vertices. These derangements are in bijection with the spheres in the wedge sum describing the homotopy type of the boolean complex for this graph. |
For each integer $t\ge 5$, we give a polynomial-time algorithm to test whether a graph contains an induced cycle with length at least $t$ and odd. |
We consider the following open question in the spirit of Ramsey theory: Given an aperiodic infinite word $w$, does there exist a finite coloring of its factors such that no factorization of $w$ is monochromatic? We show that such a coloring always exists whenever $w$ is a Sturmian word or a standard episturmian word. |
A square is a factor $S = (S_1; S_2)$ where $S_1$ and $S_2$ have the same pattern, and a permutation is said to be square-free if it contains no non-trivial squares. The permutation is further said to be bicrucial if every extension to the left or right contains a square. |
Exploiting the fact that the $q$-Whittaker polynomials arise as a specialization of the (modified) Macdonald polynomials, we derive some of their basic properties, and explore interesting identities that they satisfy. We also show how they arise as graded Frobenius characteristics of $§_n$-modules, and give a combinat... |
Inversion sequences are integer sequences $(\sigma_1, \dots, \sigma_n)$ such that $0 \leqslant \sigma_i < i$ for all $1 \leqslant i \leqslant n$. The study of pattern-avoiding inversion sequences began in two independent articles by Mansour-Shattuck and Corteel-Martinez-Savage-Weselcouch in 2015 and 2016. |
An edge-colored graph $G$ is conflict-free connected if any two of its vertices are connected by a path, which contains a color used on exactly one of its edges. The conflict-free connection number of a connected graph $G$, denoted by $cfc(G)$, is defined as the minimum number of colors that are required in order to m... |
For any complex number $\alpha$ and any even-size skew-symmetric matrix $B$, we define a generalization $\pfa{\alpha}(B)$ of the pfaffian $\pf(B)$ which we call the $\alpha$-pfaffian. The $\alpha$-pfaffian is a pfaffian analogue of the $\alpha$-determinant. |
We present a simple mechanism, which can be randomised, for constructing sparse $3$-uniform hypergraphs with strong expansion properties. These hypergraphs are constructed using Cayley graphs over $\mathbb{Z}_2^t$ and have vertex degree which is polylogarithmic in the number of vertices. |
For a partition $\nu$, let $\lambda,\mu\subseteq \nu$ be two distinct partitions such that $|\nu/\lambda|=|\nu/\mu|=1$. Butler conjectured that the divided difference $\operatorname{I}_{\lambda,\mu}[X;q,t]=(T_\lambda\widetilde{H}_\mu[X;q,t]-T_\mu\widetilde{H}_\lambda[X;q,t])/(T_\lambda-T_\mu)$ of modified Macdonald po... |
We extend two well-known results in Ramsey theory from from $K_n$ to arbitrary $n$-chromatic graphs. The first is a note of Erd\H os and Rado stating that in every 2-coloring of the edges of $K_n$ there is a monochromatic tree on $n$ vertices. |
A mixed multigraph is a multigraph which may contain both undirected and directed edges. An orientation of a mixed multigraph $G$ is an assignment of exactly one direction to each undirected edge of $G$. |
An abstract topological graph (briefly an AT-graph) is a pair $A=(G,\mathcal{X})$ where $G=(V,E)$ is a graph and $\mathcal{X}\subseteq {E \choose 2}$ is a set of pairs of its edges. The AT-graph $A$ is simply realizable if $G$ can be drawn in the plane so that each pair of edges from $\mathcal{X}$ crosses exactly once... |
A Hamilton Berge cycle of a hypergraph on $n$ vertices is an alternating sequence $(v_1, e_1, v_2, \ldots, v_n, e_n)$ of distinct vertices $v_1, \ldots, v_n$ and distinct hyperedges $e_1, \ldots, e_n$ such that $\{v_1,v_n\}\subseteq e_n$ and $\{v_i, v_{i+1}\} \subseteq e_i$ for every $i\in [n-1]$. We prove the followi... |
The scattering number of a graph $G$ was defined by Jung in 1978 as $sc(G) = max \{\omega(G - S) - |S|, S \subseteq V, \omega(G - S) \neq1\}$ where $\omega(G - S) $ is the number of connected components of the graph $G-S$. It is a measure of vulnerability of a graph and it has a direct relationship with the toughness ... |
The main result of this paper is a proof of the following conjecture of Babson & Kozlov: <br>Theorem. Let G be a graph of maximal valency d, then the complex Hom(G,K_n) is at least (n-d-2)-connected. |
Given a graph $M,$ path eigenvalues are eigenvalues of its path matrix. The path energy of a simple graph $M$ is equal to the sum of the absolute values of the path eigenvalues of the graph $M$ (Shikare et. al, 2018). |
Give a digraph $D=(V(D),A(D))$, let $\partial^+_D(v)=\{vw|w\in N^+_D(v)\}$ and $\partial^-_D(v)=\{uv|u\in N^-_D(v)\}$ be semi-cuts of $v$. A mapping $\varphi:A(D)\rightarrow [k]$ is called a weak-odd $k$-edge coloring of $D$ if it satisfies the condition: for each $v\in V(D)$, there is at least one color with an odd n... |
In this paper we introduce three combinatorial models for symmetrized poly-Bernoulli numbers. Based on our models we derive generalizations of some identities for poly-Bernoulli numbers. |
We introduce some new classes of words and permutations characterized by the second difference condition $\pi(i-1) + \pi(i+1) - 2\pi(i) \leq k$, which we call the $k$-convexity condition. We demonstrate that for any sized alphabet and convexity parameter $k$, we may find a generating function which counts $k$-convex w... |
Let $G=(V,E)$ be a finite undirected graph. Orient the edges of $G$ in an arbitrary way. |
In their classical paper, Erdős, Goodman and Pósa studied the representation of a graph with vertex set $[n]$ by a family of subsets $S_1,\dots, S_n$ with the property that $\{i,j\}$ is an edge if and only if $S_i\cap S_j\neq \emptyset$. In this note, we consider a similar representation of bounded degree $r$-uniform ... |
Motivated by integral point sets in the Euclidean plane, we consider integral point sets in affine planes over finite fields. An integral point set is a set of points in the affine plane $\mathbb{F}_q^2$ over a finite field $\mathbb{F}_q$, where the formally defined squared Euclidean distance of every pair of points i... |
This thesis is divided into three parts. The first part deals with cylindric plane partitions. |
An integer additive set-indexer is defined as an injective function $f:V(G)\rightarrow 2^{\mathbb{N}_0}$ such that the induced function $g_f:E(G) \rightarrow 2^{\mathbb{N}_0}$ defined by $g_f (uv) = f (u)+ f |
Fifty years ago Erdős asked to determine the minimum number of $k$-cliques in a graph on $n$ vertices with independence number less than l. He conjectured that this minimum is achieved by the disjoint union of $l-1$ complete graphs of size $\frac{n}{l-1}$. |
A bijective mapping $f: V(G) \rightarrow \left\{1,2,\ldots,n\right\}$ is called a \emph{Distance Magic Labeling (DML) of $G$} if ~ ${\sum_{v \in N (u)}} f |
We explore monoids generated by operators on certain infinite partial orders. Our starting point is the work of Fomin and Greene on monoids satisfying the relations $(ur+\u{r+1})\u{r+1}ur=\u{r+1}ur(ur+\u{r+1})$ and $urut=usur$ if $|r-t|>1.$ Given such a monoid, the non-commutative functions in the variables $\u{}$ ... |
A path system $\mathcal{P}$ in a graph $G=(V,E)$ is said to be irreducible if there does not exist a partition $V= A\sqcup B$ such that $\mathcal{P}$ restricts to a path system on both $G[A]$ and $G[B]$. In this paper, we construct an infinite family of non-metrizable irreducible path systems defined on certain Paley ... |
We show that a 2-subset-regular self-complementary 3-uniform hypergraph with $n$ vertices exists if and only if $n\ge 6$ and $n$ is congruent to 2 modulo 4. |
A linear code with parameters $[n, k, n - k + 1]$ is called maximum distance separable (MDS), and one with parameters $[n, k, n - k]$ is called almost MDS (AMDS). A code is near-MDS (NMDS) if both it and its dual are AMDS. |
This thesis deals with the Tutte polynomial, studied from different points of view. In the first part, we address the enumeration of planar maps equipped with a spanning forest, here called forested maps, with a weight $z$ per face and a weight $u$ per non-root component of the forest. |
Neural codes serve as a language for neurons in the brain. Convex codes, which arise from the pattern of intersections of convex sets in Euclidean space, are of particular relevance to neuroscience. |
Nestohedra are a family of convex polytopes that includes permutohedra, associahedra, and graph associahedra. In this paper, we study an extension of such polytopes, called extended nestohedra. |
Let $\sigma=(\sigma_1,..., \sigma_N)$, where $\sigma_i =\pm 1$, and let $C(\sigma)$ denote the number of permutations $\pi$ of $1,2,..., N+1,$ whose up-down signature $\mathrm{sign}(\pi(i+1)-\pi(i))=\sigma_i$, for $i=1,...,N$. <br>We prove that the set of all up-down numbers $C(\sigma)$ can be expressed by a single un... |
A graph is $1$-$planar$ if it can be drawn in the plane so that each edge is crossed by at most one other edge. Moreover, a 1-planar graph $G$ is $optimal$ if it satisfies $|E(G)|=4|V(G)|-8$. |
It is proved that all 4-edge-colourings of a (sub)cubic graph are Kempe equivalent. This resolves a conjecture of the second author. |
The active bijection forms a package of results studied by the authors in a series of papers in oriented matroids. The present paper is intended to state the main results in the particular case, and more widespread language, of graphs. |
We give some results about a bijection associating each permutation with a subexcedant function. This function is related to a particular decomposition of the permutation as a product of transpositions and therefore it has been called transposition array in the literature. |
Let ${\mathcal G}$ be a graph family defined on a common (labeled) vertex set $V$. A set $S\subseteq V$ is said to be a simultaneous metric generator for ${\cal G}$ if for every $G\in {\cal G}$ and every pair of different vertices $u,v\in V$ there exists $s\in S$ such that $d_{G}(s,u)\ne d_{G}(s,v)$, where $d_{G}$ den... |
The metric dimension dim(G) of a graph $G$ is the minimum cardinality of a subset $S$ of vertices of $G$ such that each vertex of $G$ is uniquely determined by its distances to $S$. It is well-known that the metric dimension of a graph can be drastically increased by the modification of a single edge. |
Let $\mathscr{M}$ be a conditional oriented matroid. We define a graded algebra $\widehat{\mathscr{VG}}_\mathscr{M}$ with vector space dimension given by the number of covectors in $\mathscr{M}$ which admits a distinguished filtration indexed by the poset $\mathscr{L}(\mathscr{M})$ of flats of $\mathscr{M}$. |
In a recent paper, Jason P. Bell and Jeffrey Shallit introduced the notion of {\em Lie complexity} and proved that the Lie complexity function of an automatic sequence is automatic. In this note, we give more facts concerning Lie complexity and define the extended Lie complexity and the prefix Lie complexity. |
$F$-polynomials are integer coefficient polynomials encoding the mutations of cluster variables inside a cluster algebra. In this article, we study the $F$-polynomials associated with the action of Donaldson-Thomas transformations on cluster variables. |
For the Chebyshev-Stirling numbers, a special case of the Jacobi-Stirling numbers, asymptotic formulae are derived in terms of a local central limit theorem. The underlying probabilistic approach also applies to the classical Stirling numbers of the second kind. |
The concept of $t$-difference operator for functions of partitions is introduced to prove a generalization of Stanley's theorem on polynomiality of Plancherel averages of symmetric functions related to contents and hook lengths. Our extension uses a generalization of the notion of Plancherel measure, based on walk... |
Endowed with the binary operation of set addition, the family $\mathcal P_{{\rm fin},0}(\mathbb N)$ of all finite subsets of $\mathbb N$ containing $0$ forms a monoid, with the singleton $\{0\}$ as its neutral element. <br>We show that the only non-trivial automorphism of $\mathcal P_{{\rm fin},0}(\mathbb N)$ is the i... |
Interval parking functions are a generalization of parking functions in which cars have an interval preference for their parking. We generalize this definition to parking functions with $n$ cars and $m\geq n$ parking spots, which we call interval rational parking functions and provide a formula for their enumeration. |
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