Showing papers in "Electronic Journal of Combinatorics in 2012"
TL;DR: The family of combinatorial games consists of two-player games with perfect information, no hidden information as in some card games, no chance moves and outcome restricted to (lose, win), (tie, tie) and (draw, draw) for the two players who move alternately.
Abstract: Roughly speaking, the family of combinatorial games consists of two-player games with perfect information (no hidden information as in some card games), no chance moves (no dice) and outcome restricted to (lose, win), (tie, tie) and (draw, draw) for the two players who move alternately. Tie is an end position such as in tic-tac-toe, where no player wins, whereas draw is a dynamic tie: any position from which a player has a nonlosing move, but cannot force a win. Both the easy game of Nim and the seemingly difficult chess are examples of combinatorial games. And so is go. The shorter terminology game, games is used below to designate combinatorial games.
99 citations
TL;DR: All graphs whose binomial edge ideals have a linear resolution are characterized, and it is shown that complete graphs are the only graphs with this property.
Abstract: We characterize all graphs whose binomial edge ideals have a linear resolution. Indeed, we show that complete graphs are the only graphs with this property. We also compute some graded components of the first Betti number of the binomial edge ideal of a graph with respect to the graphical terms. Finally, we give an upper bound for the Castelnuovo-Mumford regularity of the binomial edge ideal of a closed graph.
90 citations
TL;DR: The number of intervals in this lattice is proved to be $$ m+1}{n(mn+1)} {(m+1)^2 n+m\choose n-1}.
Abstract: An $m$-ballot path of size $n$ is a path on the square grid consisting of north and east steps, starting at $(0,0)$, ending at $(mn,n)$, and never going below the line $\{x=my\}$. The set of these paths can be equipped with a lattice structure, called the $m$-Tamari lattice and denoted by $\mathcal{T}_n^{(m)}$, which generalizes the usual Tamari lattice $\mathcal{T}_n$ obtained when $m=1$. We prove that the number of intervals in this lattice is $$ \frac {m+1}{n(mn+1)} {(m+1)^2 n+m\choose n-1}. $$ This formula was recently conjectured by Bergeron in connection with the study of diagonal coinvariant spaces. The case $m=1$ was proved a few years ago by Chapoton. Our proof is based on a recursive description of intervals, which translates into a functional equation satisfied by the associated generating function. The solution of this equation is an algebraic series, obtained by a guess-and-check approach. Finding a bijective proof remains an open problem.
74 citations
TL;DR: It is proved that the domination polynomial of a graph $G$ satisfies a wide range of reduction formulas and linear recurrence relations for D(G,x) for arbitrary graphs and for various special cases are shown.
Abstract: The domination polynomial $D(G,x)$ of a graph $G$ is the generating function of its dominating sets. We prove that $D(G,x)$ satisfies a wide range of reduction formulas. We show linear recurrence relations for $D(G,x)$ for arbitrary graphs and for various special cases. We give splitting formulas for $D(G,x)$ based on articulation vertices, and more generally, on splitting sets of vertices.
63 citations
TL;DR: It is proved that among the permutations of length n with i fixed points and j excedances, the number of 321-avoiding ones equals thenumber of 132- avoiding ones, for all given i,j<=n.
Abstract: Using an unprecedented technique involving diagonals of non-rational generating functions, we prove that among the permutations of length $n$ with $i$ fixed points and $j$ excedances, the number of 321-avoiding ones equals the number of 132-avoiding ones, for any given $i,j$. Our theorem generalizes a result of Robertson, Saracino and Zeilberger. Even though bijective proofs have later been found by the author jointly with Pak and with Deutsch, this paper contains the original analytic proof that was presented at FPSAC 2003.
61 citations
TL;DR: It is shown that if the largest matching in a $k-uniform hypergraph G on $n$ vertices has precisely $s$ edges, and $n>2k^2s/\log k$, then H has at most $\binom n k - \binom {n-s} k $ edges.
Abstract: We show that if the largest matching in a $k$-uniform hypergraph $G$ on $n$ vertices has precisely $s$ edges, and $n>2k^2s/\log k$, then $H$ has at most $\binom n k - \binom {n-s} k $ edges and this upper bound is achieved only for hypergraphs in which the set of edges consists of all $k$-subsets which intersect a given set of $s$ vertices.
55 citations
TL;DR: In this article, the simplicial complex of maximal fillings of a moon polyomino is shown to be a vertex-decomposable, and thus shellable, sphere, which implies a positivity result for Schubert polynomials.
Abstract: We exhibit a canonical connection between maximal $(0,1)$-fillings of a moon polyomino avoiding north-east chains of a given length and reduced pipe dreams of a certain permutation. Following this approach we show that the simplicial complex of such maximal fillings is a vertex-decomposable, and thus shellable, sphere. In particular, this implies a positivity result for Schubert polynomials. Moreover, for Ferrers shapes we construct a bijection to maximal fillings avoiding south-east chains of the same length which specializes to a bijection between $k$-triangulations of the $n$-gon and $k$-fans of Dyck paths of length $2(n-2k)$. Using this, we translate a conjectured cyclic sieving phenomenon for $k$-triangulations with rotation to the language of $k$-flagged tableaux with promotion.
49 citations
TL;DR: The theory of lecture hall partitions is used to define a generalization of the Eulerian polynomials, for each positive integer $k, that have a simple combinatorial interpretation in terms of a single statistic on generalized inversion sequences.
Abstract: We use the theory of lecture hall partitions to define a generalization of the Eulerian polynomials, for each positive integer $k$. We show that these ${1}/{k}$- Eulerian polynomials have a simple combinatorial interpretation in terms of a single statistic on generalized inversion sequences . The theory provides a geometric realization of the polynomials as the $h^*$-polynomials of $k$- lecture hall polytopes . Many of the defining relations of the Eulerian polynomials have natural ${1}/{k}$-generalizations. In fact, these properties extend to a bivariate generalization obtained by replacing ${1}/{k}$ by a continuous variable. The bivariate polynomials have appeared in the work of Carlitz, Dillon, and Roselle on Eulerian numbers of higher order and, more recently, in the theory of rook polynomials.
46 citations
TL;DR: A combinatorial proof is provided for this unexpected threefold symmetry and a large family of non-trivial equalities of the type S_{n,132}(q)=S_{ n,132)(q') is proved.
Abstract: We prove that the total number $S_{n,132}(q)$ of copies of the pattern $q$ in all 132-avoiding permutations of length $n$ is the same for $q=231$, $q=312$, or $q=213$. We provide a combinatorial proof for this unexpected threefold symmetry. We then significantly generalize this result by proving a large family of non-trivial equalities of the type $S_{n,132}(q)=S_{n,132}(q')$.
45 citations
TL;DR: In this article, it was shown that Demazure crystals are isomorphic as classical crystals to tensor products of Kirillov{Reshetikhin crystals via a canonically chosen isomorphism.
Abstract: It has previously been shown that, at least for non-exceptional Kac{Moody Lie algebras, there is a close connection between Demazure crystals and tensor products of Kirillov{Reshetikhin crystals. In particular, certain Demazure crystals are isomorphic as classical crystals to tensor products of Kirillov{Reshetikhin crystals via a canonically chosen isomorphism. Here we show that this isomorphism intertwines the natural ane grading on Demazure crystals with a combinatorially dened energy function. As a consequence, we obtain a formula of the Demazure character in terms of the energy function, which has applications to Macdonald polynomials and q-deformed Whittaker functions.
44 citations
TL;DR: This paper will give a tight minimum co-degree condition for a 4-uniform hypergraph to contain a perfect matching.
Abstract: We will give a tight minimum co-degree condition for a 4-uniform hypergraph to contain a perfect matching.
TL;DR: It is proved that in the case of 2-bootstrap percolation on the n-dimensional hypercube the maximal time the process can take to eventually infect the entire vertex set is $\lfloor \frac{n^2}{3} \rfloor$.
Abstract: Bootstrap percolation is one of the simplest cellular automata. In $r$-bootstrap percolation on a graph $G$, an infection spreads according to the following deterministic rule: infected vertices of $G$ remain infected forever and in consecutive rounds healthy vertices with at least $r$ already infected neighbours become infected. Percolation occurs if eventually every vertex is infected. In this paper we prove that in the case of $2$-bootstrap percolation on the $n$-dimensional hypercube the maximal time the process can take to eventually infect the entire vertex set is $\lfloor \frac{n^2}{3} \rfloor$.
TL;DR: The 2-color hypergraph Ramsey number of a $k$-uniform loose 3-cycle or 4-cycleHypergraph Ramsey numbers for loose cycles (and paths) were determined recently.
Abstract: Asymptotic values of hypergraph Ramsey numbers for loose cycles (and paths) were determined recently. Here we determine some of them exactly, for example the 2-color hypergraph Ramsey number of a $k$-uniform loose 3-cycle or 4-cycle: $R(\mathcal{C}^k_3,\mathcal{C}^k_3)=3k-2$ and $R(\mathcal{C}_4^k,\mathcal{C}_4^k)=4k-3$ (for $k\geq 3$). For more than 3-colors we could prove only that $R(\mathcal{C}^3_3,\mathcal{C}^3_3,\mathcal{C}^3_3)=8$. Nevertheless, the $r$-color Ramsey number of triangles for hypergraphs are much smaller than for graphs: for $r\geq 3$, $$r+5\le R(\mathcal{C}_3^3,\mathcal{C}_3^3,\dots,\mathcal{C}_3^3)\le 3r$$
TL;DR: An algebraic approach is presented and an O(C)-time algorithm is presented for computing the Roman domination numbers of special classes of graphs called polygraphs, which include rotagraphs and fasciagraphs.
Abstract: Roman domination is an historically inspired variety of domination in graphs, in which vertices are assigned a value from the set $\{0,1,2\}$ in such a way that every vertex assigned the value 0 is adjacent to a vertex assigned the value 2. The Roman domination number is the minimum possible sum of all values in such an assignment. Using an algebraic approach we present an $O(C)$-time algorithm for computing the Roman domination numbers of special classes of graphs called polygraphs, which include rotagraphs and fasciagraphs. Using this algorithm we determine formulas for the Roman domination numbers of the Cartesian products of the form $P_n\Box P_k$, $P_n\Box C_k$, for $k\leq8$ and $n \in {\mathbb N}$, and $C_n\Box P_k$ and $C_n\Box C_k$, for $k\leq 6$ and $n \in {\mathbb N}$, for paths $P_n$ and cycles $C_n$. We also find all special graphs called Roman graphs in these families of graphs.
TL;DR: It is proved that every nonnull, primitive graph with infinite diameter and countably many vertices has distinguishing number $2; so, too, is any infinite primitive permutation group with finite suborbits.
Abstract: The distinguishing number of a group $G$ acting faithfully on a set $V$ is the least number of colors needed to color the elements of $V$ so that no non-identity element of the group preserves the coloring. The distinguishing number of a graph is the distinguishing number of its automorphism group acting on its vertex set. A connected graph $\Gamma$ is said to have connectivity 1 if there exists a vertex $\alpha \in V\Gamma$ such that $\Gamma \setminus \{\alpha\}$ is not connected. For $\alpha \in V$, an orbit of the point stabilizer $G_\alpha$ is called a suborbit of $G$. We prove that every nonnull, primitive graph with infinite diameter and countably many vertices has distinguishing number $2$. Consequently, any nonnull, infinite, primitive, locally finite graph is $2$-distinguishable; so, too, is any infinite primitive permutation group with finite suborbits. We also show that all denumerable vertex-transitive graphs of connectivity 1 and all Cartesian products of connected denumerable graphs of infinite diameter have distinguishing number $2$. All of our results follow directly from a versatile lemma which we call The Distinct Spheres Lemma.
TL;DR: The Boolean algebra of integral sets are determined for dihedral group and finite abelian groups as those finite groups where the Boolean algebra generated by integral sets equals the Booleangebra generated by its subgroups.
Abstract: Integral sets of finite groups are discussed and related to the integral Cayley graphs. The Boolean algebra of integral sets are determined for dihedral group and finite abelian groups. We characterize the finite abelian groups as those finite groups where the Boolean algebra generated by integral sets equals the Boolean algebra generated by its subgroups.
TL;DR: The necessary and sufficient conditions for a regular Cayley graph to be Ramanujan were given in this article, and the spectral moments of the line graph of the graph were derived.
Abstract: Let $R$ be a finite commutative ring. The unitary Cayley graph of $R$, denoted $G_R$, is the graph with vertex set $R$ and edge set $\left\{\{a,b\}:a,b\in R, a-b\in R^\times\right\}$, where $R^\times$ is the set of units of $R$. An $r$-regular graph is Ramanujan if the absolute value of every eigenvalue of it other than $\pm r$ is at most $2\sqrt{r-1}$. In this paper we give a necessary and sufficient condition for $G_R$ to be Ramanujan, and a necessary and sufficient condition for the complement of $G_R$ to be Ramanujan. We also determine the energy of the line graph of $G_R$, and compute the spectral moments of $G_R$ and its line graph.
TL;DR: The divisibility requirement is improved and it is shown that in the above results it is enough to assume that $n$ is a multiple of $k-1$, which is best possible.
Abstract: In the random $k$-uniform hypergraph $H^{(k)}_{n,p}$ of order $n$, each possible $k$-tuple appears independently with probability $p$. A loose Hamilton cycle is a cycle of order $n$ in which every pair of consecutive edges intersects in a single vertex. It was shown by Frieze that if $p\geq c(\log n)/n^2$ for some absolute constant $c>0$, then a.a.s. $H^{(3)}_{n,p}$ contains a loose Hamilton cycle, provided that $n$ is divisible by $4$. Subsequently, Dudek and Frieze extended this result for any uniformity $k\ge 4$, proving that if $p\gg (\log n)/n^{k-1}$, then $H^{(k)}_{n,p}$ contains a loose Hamilton cycle, provided that $n$ is divisible by $2(k-1)$. In this paper, we improve the divisibility requirement and show that in the above results it is enough to assume that $n$ is a multiple of $k-1$, which is best possible.
TL;DR: This paper uses Kahn's entropy argument and Zhao's extension to prove that if G is a bipartite graph on $n$ vertices, and $\delta(G)\geq\delta$ and $\Delta(G))\leq \Delta$, then i(G)-leq i(K_{\d delta,\Delta})^{n/2 \delta}$.
Abstract: The enumeration of independent sets in graphs with various restrictions has been a topic of much interest of late. Let $i(G)$ be the number of independent sets in a graph $G$ and let $i_t(G)$ be the number of independent sets in $G$ of size $t$. Kahn used entropy to show that if $G$ is an $r$-regular bipartite graph with $n$ vertices, then $i(G)\leq i(K_{r,r})^{n/2r}$. Zhao used bipartite double covers to extend this bound to general $r$-regular graphs. Galvin proved that if $G$ is a graph with $\delta(G)\geq \delta$ and $n$ large enough, then $i(G)\leq i(K_{\delta,n-\delta})$. In this paper, we prove that if $G$ is a bipartite graph on $n$ vertices with $\delta(G)\geq\delta$ where $n\geq 2\delta$, then $i_t(G)\leq i_t(K_{\delta,n-\delta})$ when $t\geq 3$. We note that this result cannot be extended to $t=2$ (and is trivial for $t=0,1$). Also, we use Kahn's entropy argument and Zhao's extension to prove that if $G$ is a graph with $n$ vertices, $\delta(G)\geq\delta$, and $\Delta(G)\leq \Delta$, then $i(G)\leq i(K_{\delta,\Delta})^{n/2\delta}$.
TL;DR: The main aims of this paper are to obtain quantitative information about the distortion of the hyperbolicity constant of the graph G\setminus e obtained from the graph $G$ by deleting an arbitrary edge $e$ from it.
Abstract: If $X$ is a geodesic metric space and $x_1,x_2,x_3\in X$, a geodesic triangle $T=\{x_1,x_2,x_3\}$ is the union of the three geodesics $[x_1x_2]$, $[x_2x_3]$ and $[x_3x_1]$ in $X$ The space $X$ is $\delta$- hyperbolic $($in the Gromov sense$)$ if any side of $T$ is contained in a $\delta$-neighborhood of the union of the other two sides, for every geodesic triangle $T$ in $X$ We denote by $\delta(X)$ the sharp hyperbolicity constant of $X$, ie, $\delta(X):=\inf\{\delta\ge 0: \, X \, \text{ is $\delta$-hyperbolic}\,\}$ The study of hyperbolic graphs is an interesting topic since the hyperbolicity of a geodesic metric space is equivalent to the hyperbolicity of a graph related to it One of the main aims of this paper is to obtain quantitative information about the distortion of the hyperbolicity constant of the graph $G\setminus e$ obtained from the graph $G$ by deleting an arbitrary edge $e$ from it These inequalities allow to obtain the other main result of this paper, which characterizes in a quantitative way the hyperbolicity of any graph in terms of local hyperbolicity
TL;DR: The aim in this paper is to present new exact Tur a n densities for individual and finite families of $3-graphs, in many cases they are also able to give corresponding stability results, and to give some new non-induced results of a similar nature.
Abstract: If $\mathcal{F}$ is a family of graphs then the Tur a n density of $\mathcal{F}$ is determined by the minimum chromatic number of the members of $\mathcal{F}$. The situation for Tur a n densities of 3-graphs is far more complex and still very unclear. Our aim in this paper is to present new exact Tur a n densities for individual and finite families of $3$-graphs, in many cases we are also able to give corresponding stability results. As well as providing new examples of individual $3$-graphs with Tur a n densities equal to $2/9,4/9,5/9$ and $3/4$ we also give examples of irrational Tur a n densities for finite families of 3-graphs, disproving a conjecture of Chung and Graham. (Pikhurko has independently disproved this conjecture by a very different method.) A central question in this area, known as Turan's problem , is to determine the Tur a n density of $K_4^{(3)}=\{123, 124, 134, 234\}$. Tur a n conjectured that this should be $5/9$. Razborov [ On 3-hypergraphs with forbidden 4-vertex configurations in SIAM J. Disc. Math. 24 (2010), 946--963] showed that if we consider the induced Tur a n problem forbidding $K_4^{(3)}$ and $E_1$, the 3-graph with 4 vertices and a single edge, then the Tur a n density is indeed $5/9$. We give some new non-induced results of a similar nature, in particular we show that $\pi(K_4^{(3)},H)=5/9$ for a $3$-graph $H$ satisfying $\pi(H)=3/4$. We end with a number of open questions focusing mainly on the topic of which values can occur as Tur a n densities. Our work is mainly computational, making use of Razborov's flag algebra framework. However all proofs are exact in the sense that they can be verified without the use of any floating point operations. Indeed all verifying computations use only integer operations, working either over $\mathbb{Q}$ or in the case of irrational Tur a n densities over an appropriate quadratic extension of $\mathbb{Q}$.
TL;DR: A complete classification of doubly even and extremal self-dual codes of length $40$ was given in this paper, and a classification of binary extremal dual codes with length $38$ was also given.
Abstract: A complete classification of binary doubly even self-dual codes of length $40$ is given. As a consequence, a classification of binary extremal self-dual codes of length $38$ is also given.
TL;DR: It is proved that, for some constant $c=c(k,\ell)$ and sufficiently large $n$, for every coloring of the edges of $K_n^{(k)$ in which the maximum degree of the subhypergraph induced by any single color is bounded by $c'n^{k-\ell}', there exists a properly colored $\ell$-overlapping Hamilton cycle $C$ that is every two adjacent edges receive different colors.
Abstract: Let $K_n^{(k)}$ be the complete $k$-uniform hypergraph, $k\ge3$, and let $\ell$ be an integer such that $1\le \ell\le k-1$ and $k-\ell$ divides $n$. An $\ell$-overlapping Hamilton cycle in $K_n^{(k)}$ is a spanning subhypergraph $C$ of $K_n^{(k)}$ with $n/(k-\ell)$ edges and such that for some cyclic ordering of the vertices each edge of $C$ consists of $k$ consecutive vertices and every pair of adjacent edges in $C$ intersects in precisely $\ell$ vertices. We show that, for some constant $c=c(k,\ell)$ and sufficiently large $n$, for every coloring (partition) of the edges of $K_n^{(k)}$ which uses arbitrarily many colors but no color appears more than $cn^{k-\ell}$ times, there exists a rainbow $\ell$-overlapping Hamilton cycle $C$, that is every edge of $C$ receives a different color. We also prove that, for some constant $c'=c'(k,\ell)$ and sufficiently large $n$, for every coloring of the edges of $K_n^{(k)}$ in which the maximum degree of the subhypergraph induced by any single color is bounded by $c'n^{k-\ell}$, there exists a properly colored $\ell$-overlapping Hamilton cycle $C$, that is every two adjacent edges receive different colors. For $\ell=1$, both results are (trivially) best possible up to the constants. It is an open question if our results are also optimal for $2\le\ell\le k-1$. The proofs rely on a version of the Lovasz Local Lemma and incorporate some ideas from Albert, Frieze, and Reed.
TL;DR: It is proved that if for each $N \in \mathbb N$, $\Pi_N$ is a partial linear space with $N$ points and £N$ lines, then $\alpha(\ Pi_N) \gtrsim \frac{1}{e}N^{3/2}$ as “N \rightarrow \infty”.
Abstract: Let $\Pi = (P,L,I)$ denote a rank two geometry In this paper, we are interested in the largest value of $|X||Y|$ where $X \subset P$ and $Y \subset L$ are sets such that $(X \times Y) \cap I = \emptyset$ Let $\alpha(\Pi)$ denote this value We concentrate on the case where $P$ is the point set of $\mathsf{PG}(n,q)$ and $L$ is the set of $k$-spaces in $\mathsf{PG}(n,q)$ In the case that $\Pi$ is the projective plane $\mathsf{PG}(2,q)$, where $P$ is the set of points and $L$ is the set of lines of the projective plane, Haemers proved that maximal arcs in projective planes together with the set of lines not intersecting the maximal arc determine $\alpha(\mathsf{PG}(2,q))$ when $q$ is an even power of $2$ Therefore, in those cases, \[ \alpha(\Pi) = q(q - \sqrt{q} + 1)^2\] We give both a short combinatorial proof and a linear algebraic proof of this result, and consider the analogous problem in generalized polygons More generally, if $P$ is the point set of $\mathsf{PG}(n,q)$ and $L$ is the set of $k$-spaces in $\mathsf{PG}(n,q)$, where $1 \leq k \leq n - 1$, and $\Pi_q = (P,L,I)$, then we show as $q \rightarrow \infty$ that \[ \frac{1}{4}q^{(k + 2)(n - k)} \lesssim \alpha(\Pi) \lesssim q^{(k + 2)(n - k)}\] The upper bounds are proved by combinatorial and spectral techniques This leaves the open question as to the smallest possible value of $\alpha(\Pi)$ for each value of $k$ We prove that if for each $N \in \mathbb N$, $\Pi_N$ is a partial linear space with $N$ points and $N$ lines, then $\alpha(\Pi_N) \gtrsim \frac{1}{e}N^{3/2}$ as $N \rightarrow \infty$
TL;DR: The extremal tree which minimizes the total number of subtrees among the set of all $q-ary trees with $n$ non-leaf vertices is identified and the extremal $n-vertex tree with given domination number maximizing the totalNumber of subtree is characterized.
Abstract: When considering the total number of subtrees of trees, the extremal structures which maximize this number among binary trees and trees with a given maximum degree lead to some interesting facts that correlate to some other graphical indices in applications. Along this line, it is interesting to study that over some types of trees with a given order, which trees minimize or maximize this number. Here are our main results: (1) The extremal tree which minimizes the total number of subtrees among $n$-vertex trees with $k$ pendants is characterized. (2) The extremal tree which maximizes (resp. minimizes) the total number of subtrees among $n$-vertex trees with a given bipartition is characterized. (3) The extremal tree which minimizes the total number of subtrees among the set of all $q$-ary trees with $n$ non-leaf vertices is identified. (4) The extremal $n$-vertex tree with given domination number maximizing the total number of subtrees is characterized.
TL;DR: This paper finds chromatic, clique and independence number of the unitary Cayley graph of a ring $R, where $R$ is a finite ring, and proves that if $G_{R} \simeq G_{S}$, then $R\simeseq M_{n}(F)$, where R is a ring and $F$ isA finite field.
Abstract: Let $R$ be a ring with identity. The unitary Cayley graph of a ring $R$, denoted by $G_{R}$, is the graph, whose vertex set is $R$, and in which $\{x,y\}$ is an edge if and only if $x-y$ is a unit of $R$. In this paper we find chromatic, clique and independence number of $G_{R}$, where $R$ is a finite ring. Also, we prove that if $G_{R} \simeq G_{S}$, then $G_{R/J_{R}} \simeq G_{S/J_{S}}$, where $\rm J_{R}$ and $\rm J_{S}$ are Jacobson radicals of $R$ and $S$, respectively. Moreover, we prove if $G_{R} \simeq G_{M_{n}(F)}$ then $R\simeq M_{n}(F)$, where $R$ is a ring and $F$ is a finite field. Finally, let $R$ and $S$ be finite commutative rings, we show that if $G_{R} \simeq G_{S}$, then $\rm R/ {J}_{R}\simeq S/J_{S}$.
TL;DR: In this paper, the enumeration of binary trees avoiding non-contiguous binary tree patterns was studied, and it was shown that there is exactly one Wilf class of k-leaf tree patterns for any positive integer k.
Abstract: In this paper we consider the enumeration of binary trees avoiding non-contiguous binary tree patterns. We begin by computing closed formulas for the number of trees avoiding a single binary tree pattern with 4 or fewer leaves and compare these results to analogous work for contiguous tree patterns. Next, we give an explicit generating function that counts binary trees avoiding a single non-contiguous tree pattern according to number of leaves and show that there is exactly one Wilf class of k-leaf tree patterns for any positive integer k. In addition, we give a bijection between between certain sets of pattern-avoiding trees and sets of pattern-avoiding permutations. Finally, we enumerate binary trees that simultaneously avoid more than one tree pattern.
TL;DR: In this article, the authors consider the set of permutations of length 3 avoiding the pattern 132 and prove that the number of 231 patterns is the same in each permutation set.
Abstract: Each length $k$ pattern occurs equally often in the set $S_n$ of all permutations of length $n$, but the same is not true in general for a proper subset of $S_n$. Miklos Bona recently proved that if we consider the set of $n$-permutations avoiding the pattern 132, all other non-monotone patterns of length 3 are equally common. In this paper we focus on the set $\operatorname{Av}_n (123)$ of $n$-permutations avoiding $123$, and give exact formulae for the occurrences of each length 3 pattern. While this set does not have the same symmetries as $\operatorname{Av}_n (132)$, we find several similarities between the two and prove that the number of 231 patterns is the same in each.
TL;DR: An extension of Spivey's Bell number formula and its associated Bell polynomial extension is established by using Hsu-Shiue's generalized Stirling numbers and Gould-Quaintance's new Bell number formulas are extended.
Abstract: We establish an extension of Spivey's Bell number formula and its associated Bell polynomial extension by using Hsu-Shiue's generalized Stirling numbers. By means of the extension of Spivey's Bell number formula we also extend Gould-Quaintance's new Bell number formulas.
TL;DR: A generating tree approach is used to construct a recursive bijection between the set $A_{2n}(2143)$ of alternating permutations of length $2n$ avoiding $2143$ and the set of standard Young tableaux of shape $\langle n,n, n, n\rangle$.
Abstract: We extend earlier work of the same author to enumerate alternating permutations avoiding the permutation pattern $2143$. We use a generating tree approach to construct a recursive bijection between the set $A_{2n}(2143)$ of alternating permutations of length $2n$ avoiding $2143$ and the set of standard Young tableaux of shape $\langle n, n, n\rangle$, and between the set $A_{2n + 1}(2143)$ of alternating permutations of length $2n + 1$ avoiding $2143$ and the set of shifted standard Young tableaux of shape $\langle n + 2, n + 1, n\rangle$. We also give a number of conjectures and open questions on pattern avoidance in alternating permutations and generalizations thereof.