Showing papers in "Electronic Journal of Combinatorics in 2017"
TL;DR: In this article, it was shown that the clique number of a power graph is at most countably infinite in a bounded exponent group and that the power graph of a group with a constant number of vertices has finite independence number.
Abstract: Let $G$ be a group. The power graph of $G$ is a graph with the vertex set $G$, having an edge between two elements whenever one is a power of the other. We characterize nilpotent groups whose power graphs have finite independence number. For a bounded exponent group, we prove its power graph is a perfect graph and we determine its clique/chromatic number. Furthermore, it is proved that for every group $G$, the clique number of the power graph of $G$ is at most countably infinite. We also measure how close the power graph is to the commuting graph by introducing a new graph which lies in between. We call this new graph as the enhanced power graph . For an arbitrary pair of these three graphs we characterize finite groups for which this pair of graphs are equal.
66 citations
TL;DR: The Strong Spectral Property and the Strong Multiplicity Property as discussed by the authors are extensions of the Strong Arnold Property that target a better understanding of all possible spectra and their associated multiplicities.
Abstract: For a given graph $G$ and an associated class of real symmetric matrices whose diagonal entries are governed by the adjacencies in $G$, the collection of all possible spectra for such matrices is considered. Building on the pioneering work of Colin de Verdi e re in connection with the Strong Arnold Property, two extensions are devised that target a better understanding of all possible spectra and their associated multiplicities. These new properties are referred to as the Strong Spectral Property and the Strong Multiplicity Property. Finally, these ideas are applied to the minimum number of distinct eigenvalues associated with $G$, denoted by $q(G)$. The graphs for which $q(G)$ is at least the number of vertices of $G$ less one are characterized.
39 citations
TL;DR: In this article, the authors presented new results concerning threshold functions for a wide family of random intersection graphs and improved and generalized the coupling method introduced for RIGs so that it may be used for a wider range of parameters.
Abstract: We present new results concerning threshold functions for a wide family of random intersection graphs. To this end we improve and generalize the coupling method introduced for random intersection graphs so that it may be used for a wider range of parameters. Using the new approach we are able to tighten the best known results concerning random intersection graphs and establish threshold functions for some monotone properties of inhomogeneous random intersection graphs. Considered properties are: $k$-connectivity, matching containment and hamiltonicity.
35 citations
TL;DR: In this paper, it was shown that for each k ≥ 0, a tree with tree-width at most k and queue-number at most 2^k-1 can be constructed with a track-number of at most O(k^2) for any k = 2.
Abstract: A queue layout of a graph consists of a linear order on the vertices and an assignment of the edges to queues , such that no two edges in a single queue are nested. The minimum number of queues needed in a queue layout of a graph is called its queue-number . We show that for each $k\geq0$, graphs with tree-width at most $k$ have queue-number at most $2^k-1$. This improves upon double exponential upper bounds due to Dujmovic et al. and Giacomo et al. As a consequence we obtain that these graphs have track-number at most $2^{O(k^2)}$. We complement these results by a construction of $k$-trees that have queue-number at least $k+1$. Already in the case $k=2$ this is an improvement to existing results and solves a problem of Rengarajan and Veni Madhavan, namely, that the maximal queue-number of $2$-trees is equal to $3$.
33 citations
TL;DR: This algorithm was designed by a computer program that searched for optimal algorithms for small values of $d by means of a randomised distributed algorithm: each node needs to produce only one random bit, and the algorithm runs in one synchronous communication round.
Abstract: Let $G$ be a $d$-regular triangle-free graph with $m$ edges. We present an algorithm which finds a cut in $G$ with at least $(1/2 + 0.28125/\sqrt{d})m$ edges in expectation, improving upon Shearer's classic result. In particular, this implies that any $d$-regular triangle-free graph has a cut of at least this size, and thus, we obtain a new lower bound for the maximum number of edges in a bipartite subgraph of $G$. Our algorithm is simpler than Shearer's classic algorithm and it can be interpreted as a very efficient randomised distributed (local) algorithm : each node needs to produce only one random bit, and the algorithm runs in one round. The randomised algorithm itself was discovered using computational techniques . We show that for any fixed $d$, there exists a weighted neighbourhood graph $\mathcal{N}_d$ such that there is a one-to-one correspondence between heavy cuts of $\mathcal{N}_d$ and randomised local algorithms that find large cuts in any $d$-regular input graph. This turns out to be a useful tool for analysing the existence of cuts in $d$-regular graphs: we can compute the optimal cut of $\mathcal{N}_d$ to attain a lower bound on the maximum cut size of any $d$-regular triangle-free graph.
33 citations
TL;DR: In this paper, infinite families of non-linear maximum rank distance codes are constructed by using the setting of bilinear forms of a finite vector space and it is shown that these families contain the non- linear maximum rankdistance codes recently provided by Cossidente, Marino and Pavese.
Abstract: In this paper we construct infinite families of non-linear maximum rank distance codes by using the setting of bilinear forms of a finite vector space. We also give a geometric description of such codes by using the cyclic model for the field reduction of finite geometries and we show that these families contain the non-linear maximum rank distance codes recently provided by Cossidente, Marino and Pavese.
33 citations
TL;DR: In this paper, it was shown that a partition into monochromatic components is possible for sufficiently large complete graphs with large minimum degree, where the components are only required to form a cover.
Abstract: Erdős , Gyarfas , and Pyber (1991) conjectured that every $r$ -colored complete graph can be partitioned into at most $r-1$ monochromatic components; this is a strengthening of a conjecture of Lov asz (1975) and Ryser (1970) in which the components are only required to form a cover. An important partial result of Haxell and Kohayakawa (1995) shows that a partition into $r$ monochromatic components is possible for sufficiently large $r$ -colored complete graphs. We start by extending Haxell and Kohayakawa's result to graphs with large minimum degree, then we provide some partial analogs of their result for random graphs. In particular, we show that if $p\ge \left(\frac{27\log n}{n}\right)^{1/3}$ , then a.a.s. in every $2$ -coloring of $G(n,p)$ there exists a partition into two monochromatic components, and for $r\geq 2$ if $p\ll \left(\frac{r\log n}{n}\right)^{1/r}$ , then a.a.s. there exists an $r$ -coloring of $G(n,p)$ such that there does not exist a cover with a bounded number of components. Finally, we consider a random graph version of a classic result of Gyarfa s (1977) about large monochromatic components in $r$ -colored complete graphs. We show that if $p=\frac{\omega(1)}{n}$ , then a.a.s. in every $r$ -coloring of $G(n,p)$ there exists a monochromatic component of order at least $(1-o(1))\frac{n}{r-1}$ .
25 citations
TL;DR: The Kazhdan-Lusztig polynomial of a rank $n+1$ thagomizer matroid is computed by showing that the coefficient of $t^k$ is equal to the number of Dyck paths of semilength $n$ with long ascents.
Abstract: We introduce thagomizer matroids and compute the Kazhdan-Lusztig polynomial of a rank $n+1$ thagomizer matroid by showing that the coefficient of $t^k$ is equal to the number of Dyck paths of semilength $n$ with $k$ long ascents. We also give a conjecture for the $S_n$-equivariant Kazhdan-Lusztig polynomial of a thagomizer matroid.
25 citations
TL;DR: It is proved that a winning strategy exists if and only if $N$ is divisible by $3$ or $N=4$ and a predetermined guessing strategy is winning if it guarantees at least one correct individual guess for every assignment of colours.
Abstract: We study a cooperative game in which each member of a team of N players, wearing coloured hats and situated at the vertices of the cycle graph with N vertices, is guessing their own hat colour merely on the basis of observing the hats worn by their two neighbours without exchanging the information. Each hat can have one of three colours. A predetermined guessing strategy is winning if it guarantees at least one correct individual guess for every assignment of colours. We prove that a winning strategy exists if and only if N is divisible by 3 or N = 4. This asymmetric game is an example of relational system using incomplete information about an unpredictable situation, where at least one participant has to act properly.
23 citations
TL;DR: In this paper, it was shown that the cycle of an integral circulant graph with a cycle admits state transfer if and only if the cycle is a power of two and it occurs between every pair of antipodal vertices.
Abstract: Let $G$ be a graph with adjacency matrix $A$. The transition matrix of $G$ relative to $A$ is defined by $H(t):=\exp{\left(-itA\right)}$, where $t\in {\mathbb R}$. The graph $G$ is said to admit pretty good state transfer between a pair of vertices $u$ and $v$ if there exists a sequence of real numbers $\{t_k\}$ and a complex number $\gamma$ of unit modulus such that $\lim\limits_{k\rightarrow\infty} H(t_k) e_u=\gamma e_v.$ We find that the cycle $C_n$ as well as its complement $\overline{C}_n$ admit pretty good state transfer if and only if $n$ is a power of two, and it occurs between every pair of antipodal vertices. In addition, we look for pretty good state transfer in more general circulant graphs. We prove that union (edge disjoint) of an integral circulant graph with a cycle, each on $2^k$ $(k\geq 3)$ vertices, admits pretty good state transfer. The complement of such union also admits pretty good state transfer. Using Cartesian products, we find some non-circulant graphs admitting pretty good state transfer.
23 citations
TL;DR: The maximum number of edges in a properly edge-colored graph on n vertices is the {\emph rainbow Tur\'an number} of F, and bounds are given on this maximum, disproving a conjecture in Keevash et al.
Abstract: For a fixed graph $F$, we would like to determine the maximum number of edges in a properly edge-colored graph on $n$ vertices which does not contain a rainbow copy of $F$, that is, a copy of $F$ all of whose edges receive a different color. This maximum, denoted by $ex^*(n,F)$, is the rainbow Tur a n number of $F$, and its systematic study was initiated by Keevash, Mubayi, Sudakov and Verstra e te [ Combinatorics, Probability and Computing 16 (2007)]. We determine $ex^*(n,F)$ exactly when $F$ is a forest of stars, and give bounds on $ex^*(n,F)$ when $F$ is a path with $l$ edges, disproving a conjecture in the aforementioned paper for $l=4$.
TL;DR: In this paper, it was shown that every biregular bipartite graph admits an antimagic orientation, where the labels of all arcs entering a vertex are subtracted from the labels leaving it.
Abstract: An antimagic labeling of a directed graph $D$ with $n$ vertices and $m$ arcs is a bijection from the set of arcs of $D$ to the integers $\{1, \cdots, m\}$ such that all $n$ oriented vertex sums are pairwise distinct, where an oriented vertex sum is the sum of labels of all arcs entering that vertex minus the sum of labels of all arcs leaving it. An undirected graph $G$ is said to have an antimagic orientation if $G$ has an orientation which admits an antimagic labeling. Hefetz, Mutze, and Schwartz conjectured that every connected undirected graph admits an antimagic orientation. In this paper, we support this conjecture by proving that every biregular bipartite graph admits an antimagic orientation.
TL;DR: For any finite set $A \subset \mathbb R$, the following six-variable expander results are proved.
Abstract: We prove several expanders with exponent strictly greater than $2$. For any finite set $A \subset \mathbb R$, we prove the following six-variable expander results: $|(A-A)(A-A)(A-A)| \gg \frac{|A|^{2+\frac{1}{8}}}{\log^{\frac{17}{16}}|A|},$ $\left|\frac{A+A}{A+A}+\frac{A}{A}\right| \gg \frac{|A|^{2+\frac{2}{17}}}{\log^{\frac{16}{17}}|A|},$ $\left|\frac{AA+AA}{A+A}\right| \gg \frac{|A|^{2+\frac{1}{8}}}{\log |A|},$ $\left|\frac{AA+A}{AA+A}\right| \gg \frac{|A|^{2+\frac{1}{8}}}{\log |A|}.$
TL;DR: In this article, the existence of a bijection between edge-colored graphs and some colored combinatorial maps, called stuffed Walsh maps, has been proved, which generalizes Walsh's representation of hypermaps as bipartite maps, by replacing the vertices which correspond to hyperedges with non-properly-edge-colored maps.
Abstract: Regular edge-colored graphs encode colored triangulations of pseudo-manifolds. Here we study families of edge-colored graphs built from a finite but arbitrary set of building blocks, which extend the notion of $p$-angulations to arbitrary dimensions. We prove the existence of a bijection between any such family and some colored combinatorial maps which we call stuffed Walsh maps. Those maps generalize Walsh's representation of hypermaps as bipartite maps, by replacing the vertices which correspond to hyperedges with non-properly-edge-colored maps. This shows the equivalence of tensor models with multi-trace, multi-matrix models by extending the intermediate field method perturbatively to any model. We further use the bijection to study the graphs which maximize the number of faces at fixed number of vertices and provide examples where the corresponding stuffed Walsh maps can be completely characterized.
TL;DR: This work explores conditions under which the corona of graphs G \circ H exhibits state transfer and describes new families of graphs with state transfer based on the Frucht-Harary corona product.
Abstract: We study state transfer in quantum walks on graphs relative to the adjacency matrix Our motivation is to understand how the addition of pendant subgraphs affect state transfer For two graphs $G$ and $H$, the Frucht-Harary corona product $G \circ H$ is obtained by taking $|G|$ copies of the cone $K_{1} + H$ and by identifying the conical vertices according to $G$ Our work explores conditions under which the corona $G \circ H$ exhibits state transfer We also describe new families of graphs with state transfer based on the corona product Some of these constructions provide a generalization of related known results
TL;DR: This paper counts Stirling permutations of the second kind by their cycle ascent plateaus, fixed points and cycles and presents constructive proofs of a kind of combinatorial expansions of the Eulerian polynomials of types $A$ and $B$.
Abstract: In this paper, we introduce Stirling permutations of the second kind. In particular, we count Stirling permutations of the second kind by their cycle ascent plateaus, fixed points and cycles. Moreover, we get an expansion of the ordinary derangement polynomials in terms of the Stirling derangement polynomials. Finally, we present constructive proofs of a kind of combinatorial expansions of the Eulerian polynomials of types $A$ and $B$.
TL;DR: In this article, a discrete Morse matching for a family of independence complexes for the 2-times n-grid graph is described, and the dimensions of the chain spaces for the resulting Morse complexes are derived.
Abstract: The topology of the matching complex for the $2\times n$ grid graph is mysterious. We describe a discrete Morse matching for a family of independence complexes $\mathrm{Ind}(\Delta_n^m)$ that include these matching complexes. Using this matching, we determine the dimensions of the chain spaces for the resulting Morse complexes and derive bounds on the location of non-trivial homology groups for certain $\mathrm{Ind}(\Delta_n^m)$. Further, we determine the Euler characteristic of $\mathrm{Ind}(\Delta_n^m)$ and prove that several homology groups of $\mathrm{Ind}(\Delta_n^m)$ are non-zero.
TL;DR: This variant of the chip-firing process on the infinite path graph $\mathbb{Z}$ is investigated: rather than treating the chips as indistinguishable, they are labelled with positive integers, which exhibits a remarkable confluence property, similar to but subtler than the confluence that prevails for unlabeled chip-Firing.
Abstract: We investigate a variant of the chip-firing process on the infinite path graph $\mathbb{Z}$: rather than treating the chips as indistinguishable, we label them with positive integers. To fire an unstable vertex, i.e. a vertex with more than one chip, we choose any two chips at that vertex and move the lesser-labeled chip to the left and the greater-labeled chip to the right. This labeled version of the chip-firing process exhibits a remarkable confluence property, similar to but subtler than the confluence that prevails for unlabeled chip-firing: when all chips start at the origin and the number of chips is even, the chips always end up in sorted order. Our proof of sorting relies upon an independently interesting lemma concerning unlabeled chip-firing which says that stabilization preserves a natural partial order on configurations. We also discuss some extensions of this sorting phenomenon to other graphs (variants of the infinite path), to other initial configurations, and to other Cartan-Killing types.
TL;DR: It is proved that the triple of classical permutation statistics (exc+1, den, inv — exc) on $321$-avoiding permutations is equidistributed with the triple (bk, rb, rs) on £1212- avoidanceing RGFs, which generalizes another result of Simion.
Abstract: Restricted growth functions (RGFs) avoiding the pattern $1212$ are in natural bijection with noncrossing partitions. Motivated by recent work of Campbell et al., we study five classical statistics bk, ls, lb, rs and rb on $1212$-avoiding RGFs. We show the equidistribution of (ls, rb, lb, bk) and (rb, ls, lb, bk) on $1212$-avoiding RGFs by constructing a simple involution. To our surprise, this result was already proved by Simion 22 years ago via an involution on noncrossing partitions. Our involution, though turns out essentially the same as Simion's, is defined quite differently and has the advantage that makes the discussion more transparent. Consequently, a multiset-valued extension of Simion's result is discovered. Furthermore, similar approach enables us to prove the equidistribution of (mak, rb, rs, bk) and (rb, mak, rs, bk) on $1212$-avoiding RGFs, where "mak" is a set partition statistic introduced by Steingr i msson. Through two bijections to Motzkin paths, we also prove that the triple of classical permutation statistics (exc+1, den, inv — exc) on $321$-avoiding permutations is equidistributed with the triple (bk, rb, rs) on $1212$-avoiding RGFs, which generalizes another result of Simion. In the course, an interesting $q$-analog of the $\gamma$-positivity of Narayana polynomials is found.
TL;DR: In this article, the authors analyse properties of geometric intersection graphs to show the strict containment between some natural classes of intersection graphs and show that a graph is outerplanar if and only if the 1-subdivision of the graph is an outer-segment.
Abstract: We analyse properties of geometric intersection graphs to show the strict containment between some natural classes of geometric intersection graphs. In particular, we show the following properties: A graph $G$ is outerplanar if and only if the 1-subdivision of $G$ is outer-segment. For each integer $k\ge 1$, the class of intersection graphs of segments with $k$ different lengths is a strict subclass of the class of intersection graphs of segments with $k+1$ different lengths. For each integer $k\ge 1$, the class of intersection graphs of disks with $k$ different sizes is a strict subclass of the class of intersection graphs of disks with $k+1$ different sizes. The class of outer-segment graphs is a strict subclass of the class of outer-string graphs.
TL;DR: It is proved that this inertial bound is not a lower bound for the vector chromatic number and investigated extremal graphs for these bounds and concluded with a discussion of asymmetry between $n+$ and $n^-$, including some Nordhaus-Gaddum bounds for inertia.
Abstract: Let $\chi(G$) and $\chi_f(G)$ denote the chromatic and fractional chromatic numbers of a graph $G$, and let $(n^+ , n^0 , n^-)$ denote the inertia of $G$. We prove that: \[ 1 + \max\left(\frac{n^+}{n^-} , \frac{n^-}{n^+}\right) \le \chi(G)\] and conjecture that \[ 1 + \max\left(\frac{n^+}{n^-} , \frac{n^-}{n^+}\right) \le \chi_f(G).\] We investigate extremal graphs for these bounds and demonstrate that this inertial bound is not a lower bound for the vector chromatic number. We conclude with a discussion of asymmetry between $n^+$ and $n^-$, including some Nordhaus-Gaddum bounds for inertia.
TL;DR: It is shown that for larger values of $k$ the situation is different, graphs of minimum degree can replace complete graphs and still there is a monochromatic connected component of order at least ${n\over k-1}$, in fact $$delta(G)\ge \left(1 - \frac{1}{1000(k-1)^9}\right)n$$ suffices.
Abstract: It is well-known that in every $k$-coloring of the edges of the complete graph $K_n$ there is a monochromatic connected component of order at least ${n\over k-1}$. In this paper we study an extension of this problem by replacing complete graphs by graphs of large minimum degree. For $k=2$ the authors proved that $\delta(G)\ge{3n\over 4}$ ensures a monochromatic connected component with at least $\delta(G)+1$ vertices in every $2$-coloring of the edges of a graph $G$ with $n$ vertices. This result is sharp, thus for $k=2$ we really need a complete graph to guarantee that one of the colors has a monochromatic connected spanning subgraph. Our main result here is that for larger values of $k$ the situation is different, graphs of minimum degree $(1-\epsilon_k)n$ can replace complete graphs and still there is a monochromatic connected component of order at least ${n\over k-1}$, in fact $$\delta(G)\ge \left(1 - \frac{1}{1000(k-1)^9}\right)n$$ suffices. Our second result is an improvement of this bound for $k=3$. If the edges of $G$ with $\delta(G)\geq {9n\over 10}$ are $3$-colored, then there is a monochromatic component of order at least ${n\over 2}$. We conjecture that this can be improved to ${7n\over 9}$ and for general $k$ we conjecture the following: if $k\geq 3$ and $G$ is a graph of order $n$ such that $\delta(G)\geq \left( 1 - \frac{k-1}{k^2}\right)n$, then in any $k$-coloring of the edges of $G$ there is a monochromatic connected component of order at least ${n\over k-1}$.
TL;DR: In this article, the authors provided a crystal theoretic explanation of this fact and gave an explicit combinatorial description of the coefficients in the Schur expansion in terms of highest weight crystal elements.
Abstract: Combining results of T.K. Lam and J. Stembridge, the type $C$ Stanley symmetric function $F_w^C(\mathbf{x})$, indexed by an element $w$ in the type $C$ Coxeter group, has a nonnegative integer expansion in terms of Schur functions. We provide a crystal theoretic explanation of this fact and give an explicit combinatorial description of the coefficients in the Schur expansion in terms of highest weight crystal elements.
TL;DR: In this article, the threshold for percolation was determined up to a constant factor, in the case where both graphs are Erdő s-R e nyi random graphs.
Abstract: Jigsaw percolation is a model for the process of solving puzzles within a social network, which was recently proposed by Brummitt, Chatterjee, Dey and Sivakoff. In the model there are two graphs on a single vertex set (the `people' graph and the `puzzle' graph), and vertices merge to form components if they are joined by an edge of each graph. These components then merge to form larger components if again there is an edge of each graph joining them, and so on. Percolation is said to occur if the process terminates with a single component containing every vertex. In this note we determine the threshold for percolation up to a constant factor, in the case where both graphs are Erd ő s-R e nyi random graphs.
TL;DR: In this paper, the topology of the 3-sphere of the octahedra has been characterized and a family of plane trees has been shown to have the same topology.
Abstract: 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 is suitable for combinatorics and enumeration. In particular, universality classes of three-dimensional triangulations can be investigated within this framework. Here we study colored triangulations obtained by gluing octahedra. Those which maximize the number of edges at fixed number of octahedra are fully characterized and are shown to have the topology of the 3-sphere. They are further shown to be in bijection with a family of plane trees. The enumeration is performed both directly and using this bijection.
TL;DR: The number of species of transversal-free latin squares is shown to be at least at least $n^{n^{3/2}(1/2-o(1))}$ for even $n\rightarrow\infty$.
Abstract: A $k$-plex in a latin square of order $n$ is a selection of $kn$ entries that includes $k$ representatives from each row and column and $k$ occurrences of each symbol. A $1$-plex is also known as a transversal. It is well known that if $n$ is even then $B_n$, the addition table for the integers modulo $n$, possesses no transversals. We show that there are a great many latin squares that are similar to $B_n$ and have no transversal. As a consequence, the number of species of transversal-free latin squares is shown to be at least $n^{n^{3/2}(1/2-o(1))}$ for even $n\rightarrow\infty$. We also produce various constructions for latin squares that have no transversal but do have a $k$-plex for some odd $k>1$. We prove a 2002 conjecture of the second author that for all even orders $n>4$ there is a latin square of order $n$ that contains a $3$-plex but no transversal. We also show that for odd $k$ and $m\geq 2$, there exists a latin square of order $2km$ with a $k$-plex but no $k'$-plex for odd $k'
TL;DR: In this article, the Tur a n numbers for 3-uniform hypergraphs were analyzed under some additional restrictions, and the largest number of edges in an n-vertex 3-graph which is not a star was determined.
Abstract: Let $P$ denote a 3-uniform hypergraph consisting of 7 vertices $a,b,c,d,e,f,g$ and 3 edges $\{a,b,c\}, \{c,d,e\},$ and $\{e,f,g\}$. It is known that the $r$-colored Ramsey number for $P$ is $R(P;r)=r+6$ for $r=2,3$, and that $R(P;r)\le 3r$ for all $r\ge3$. The latter result follows by a standard application of the Tur a n number $\mathrm{ex}_3(n;P)$, which was determined to be $\binom{n-1}2$ in our previous work. We have also shown that the full star is the only extremal 3-graph for $P$. In this paper, we perform a subtle analysis of the Tur a n numbers for $P$ under some additional restrictions. Most importantly, we determine the largest number of edges in an $n$-vertex $P$-free 3-graph which is not a star. These Tur a n-type results, in turn, allow us to confirm the formula $R(P;r)=r+6$ for $r\in\{4,5,6,7\}$.
TL;DR: In this paper, an elliptic rook number for Ferrers boards was constructed using elliptic weights, which generalizes Garsia and Remmel's $q$-rook numbers by two additional independent parameters $a$ and $b$ and a nome $p$.
Abstract: Utilizing elliptic weights, we construct an elliptic analogue of rook numbers for Ferrers boards. Our elliptic rook numbers generalize Garsia and Remmel's $q$-rook numbers by two additional independent parameters $a$ and $b$, and a nome $p$. The elliptic rook numbers are shown to satisfy an elliptic extension of a factorization theorem which in the classical case was established by Goldman, Joichi and White and extended to the $q$-case by Garsia and Remmel. We obtain similar results for elliptic analogues of Garsia and Remmel's $q$-file numbers for skyline boards. We also provide an elliptic extension of the $j$-attacking model introduced by Remmel and Wachs. Various applications of our results include elliptic analogues of (generalized) Stirling numbers of the first and second kind, Lah numbers, Abel numbers, and $r$-restricted versions thereof.
TL;DR: The central idea is that the $ms$-abaci of maximal $(s,ms\pm1)$-cores can be built up from $s$- abaci of $s,s\pm 1)$ -cores in an elegant way.
Abstract: We develop a geometric approach to the study of $(s,ms-1)$-core and $(s,ms+1)$-core partitions through the associated $ms$-abaci. This perspective yields new proofs for results of H. Xiong and A. Straub on the enumeration of $(s, s+1)$ and $(s,ms-1)$-core partitions with distinct parts. It also enumerates $(s, ms+1)$-cores with distinct parts. Furthermore, we calculate the weight of the $(s, ms-1,ms+1)$-core partition with the largest number of parts. Finally we use 2-core partitions to enumerate self-conjugate core partitions with distinct parts. The central idea is that the $ms$-abaci of maximal $(s,ms\pm1)$-cores can be built up from $s$-abaci of $(s,s\pm 1)$-cores in an elegant way.
TL;DR: Two results are proved on stacked triangulated manifolds: if $\Delta$ is a tight connected closed homology $d$-manifold whosei$th homology vanishes for $1 < i < d-1$, then $\Delta $ is a stacked triangling of a manifold.
Abstract: We prove two results on stacked triangulated manifolds in this paper: (a) every stacked triangulation of a connected manifold with or without boundary is obtained from a simplex or the boundary of a simplex by certain combinatorial operations; (b) in dimension $d \geq 4$, if $\Delta$ is a tight connected closed homology $d$-manifold whose $i$th homology vanishes for $1 < i < d-1$, then $\Delta$ is a stacked triangulation of a manifold. These results give affirmative answers to questions posed by Novik and Swartz and by Effenberger.