Whitney Numbers of the Second Kind for the Star Poset

The integers W0, ..., Wt are called Whitney numbers of the second kind for a ranked poset if Wk is the number of elements of rank k. The set of transpositions T = {(1, n), (2, n), ..., (n - 1, n)} generates Sn, the symmetric group. We define the star poset, a ranked poset the elements of which are those of Sn and the partial order of which is obtained from the Cayley graph using T. We characterize minimal factorizations of elements of Sn as products of generators in T and provide recurrences, generating functions and explicit formulae for the Whitney numbers of the second kind for the star poset

Decorated hypertrees

C. Jensen, J. McCammond and J. Meier have used weighted hypertrees to compute the Euler characteristic of a subgroup of the automorphism group of a free product. Weighted hypertrees also appear in the study of the homology of the hypertree poset. We link them to decorated hypertrees after a general study on decorated hypertrees, which we enumerate using box trees.---C. Jensen, J. McCammond et J. Meier ont utilis\'e des hyperarbres pond\'er\'es pour calculer la caract\'eristique d'Euler d'un sous-groupe du groupe des automorphismes d'un produit libre. Un autre type d'hyperarbres pond\'er\'es appara\^it aussi dans l'\'etude de l'homologie du poset des hyperarbres. Nous \'etudions les hyperarbres d\'ecor\'es puis les comptons \`a l'aide de la notion d'arbre en bo\^ite avant de les relier aux hyperarbres pond\'er\'es.Comment: nombre de pages : 3

A new graph invariant arises in toric topology

In this paper, we introduce new combinatorial invariants of any finite simple graph, which arise in toric topology. We compute the $i$-th (rational) Betti number and Euler characteristic of the real toric variety associated to a graph associahedron P_{\B(G)}. They can be calculated by a purely combinatorial method (in terms of graphs) and are named $a_i(G)$ and $b(G)$, respectively. To our surprise, for specific families of the graph $G$, our invariants are deeply related to well-known combinatorial sequences such as the Catalan numbers and Euler zigzag numbers.Comment: 21 pages, 3 figures, 4 table

The equivariant topology of stable Kneser graphs

The stable Kneser graph $SG_{n,k}$, $n\ge1$, $k\ge0$, introduced by Schrijver \cite{schrijver}, is a vertex critical graph with chromatic number $k+2$, its vertices are certain subsets of a set of cardinality $m=2n+k$. Bj\"orner and de Longueville \cite{anders-mark} have shown that its box complex is homotopy equivalent to a sphere, \Hom(K_2,SG_{n,k})\homot\Sphere^k. The dihedral group $D_{2m}$ acts canonically on $SG_{n,k}$, the group $C_2$ with 2 elements acts on $K_2$. We almost determine the $(C_2\times D_{2m})$-homotopy type of \Hom(K_2,SG_{n,k}) and use this to prove the following results. The graphs $SG_{2s,4}$ are homotopy test graphs, i.e. for every graph $H$ and $r\ge0$ such that \Hom(SG_{2s,4},H) is $(r-1)$-connected, the chromatic number $\chi(H)$ is at least $r+6$. If $k\notin\set{0,1,2,4,8}$ and $n\ge N(k)$ then $SG_{n,k}$ is not a homotopy test graph, i.e.\ there are a graph $G$ and an $r\ge1$ such that \Hom(SG_{n,k}, G) is $(r-1)$-connected and $\chi(G)<r+k+2$.Comment: 34 pp

Note on Ward-Horadam H(x) - binomials' recurrences and related interpretations, II

We deliver here second new $\textit{H(x)}-binomials'$ recurrence formula, were $H(x)-binomials'$ array is appointed by $Ward-Horadam$ sequence of functions which in predominantly considered cases where chosen to be polynomials . Secondly, we supply a review of selected related combinatorial interpretations of generalized binomial coefficients. We then propose also a kind of transfer of interpretation of $p,q-binomial$ coefficients onto $q-binomial$ coefficients interpretations thus bringing us back to $Gy{\"{o}}rgy P\'olya$ and Donald Ervin Knuth relevant investigation decades ago.Comment: 57 pages, 8 figure

Combinatorics of embeddings

We offer the following explanation of the statement of the Kuratowski graph planarity criterion and of 6/7 of the statement of the Robertson-Seymour-Thomas intrinsic linking criterion. Let us call a cell complex 'dichotomial' if to every cell there corresponds a unique cell with the complementary set of vertices. Then every dichotomial cell complex is PL homeomorphic to a sphere; there exist precisely two 3-dimensional dichotomial cell complexes, and their 1-skeleta are K_5 and K_{3,3}; and precisely six 4-dimensional ones, and their 1-skeleta all but one graphs of the Petersen family. In higher dimensions n>2, we observe that in order to characterize those compact n-polyhedra that embed in S^{2n} in terms of finitely many "prohibited minors", it suffices to establish finiteness of the list of all (n-1)-connected n-dimensional finite cell complexes that do not embed in S^{2n} yet all their proper subcomplexes and proper cell-like combinatorial quotients embed there. Our main result is that this list contains the n-skeleta of (2n+1)-dimensional dichotomial cell complexes. The 2-skeleta of 5-dimensional dichotomial cell complexes include (apart from the three joins of the i-skeleta of (2i+2)-simplices) at least ten non-simplicial complexes.Comment: 49 pages, 1 figure. Minor improvements in v2 (subsection 4.C on transforms of dichotomial spheres reworked to include more details; subsection 2.D "Algorithmic issues" added, etc