arXiv subjects
Publications and source records attributed to John Shareshian.
In recent years we have worked on a project involving poset topology, various analogues of Eulerian polynomials, and a refinement of Richard Stanley's chromatic symmetric function. Here we discuss how Stanley's ideas and results have influenced and inspired our own work.
We show that the order complex of the poset of all cosets of all proper subgroups of a finite group $G$ is never $\mathbb{F}_{2}$-acyclic and therefore never contractible. This settles a question of K. S. Brown.
Let p be an odd prime and let P be a p-group. We examine the order complex of the poset of elementary abelian subgroups of P having order at least p^2. S. Bouc and J. Thévenaz showed that this complex has the homotopy type of a wedge of spheres. We show that, for each nonnegative integer l, the number of spheres of dimension l in this wedge is controlled by the number of extraspecial subgroups X of P having order p^{2l+3} and satisfying Omega_1(C_P(X))=Z(X). We go on to provide a negative answer to a question raised by Bouc and Thévenaz concerning restrictions on the homology groups of the given complex.
We introduce a homological approach to exhibiting instances of Stembridge's q=-1 phenomenon. This approach is shown to explain two important instances of the phenomenon, namely that of partitions whose Ferrers diagrams fit in a rectangle of fixed size and that of plane partitions fitting in a box of fixed size. A more general framework of invariant and coinvariant complexes with coefficients taken mod 2 is developed, and as a part of this story an analogous homological result for necklaces is conjectured.
We discuss three distinct topics of independent interest; one in enumerative combinatorics, one in symmetric function theory, and one in algebraic geometry. The topic in enumerative combinatorics concerns a q-analog of a generalization of the Eulerian polynomials, the one in symmetric function theory deals with a refinement of the chromatic symmetric functions of Stanley, and the one in algebraic geometry deals with Tymoczko's representation of the symmetric group on the cohomology of the regular semisimple Hessenberg variety of type A. Our purpose is to explore some remarkable connections between these topics.
We use the theory of lexicographic shellability to provide various examples in which the rank of the homology of a Rees product of two partially ordered sets enumerates some set of combinatorial objects, perhaps according to some natural statistic on the set. Many of these examples generalize a result of J. Jonsson, which says that the rank of the unique nontrivial homology group of the Rees product of a truncated Boolean algebra of degree $n$ and a chain of length $n-1$ is the number of derangements in $§_n$.\
We show that if G is a finite group then no chain of modular elements in its subgroup lattice L(G) is longer than a chief series. Also, we show that if G is a nonsolvable finite group then every maximal chain in L(G) has length at least two more than that of the chief length of G, thereby providing a converse of a result of J. Kohler. Our results enable us to give a new characterization of finite solvable groups involving only the combinatorics of subgroup lattices. Namely, a finite group G is solvable if and only if L(G) contains a maximal chain X and a chain M consisting entirely of modular elements, such that X and M have the same length.
We introduce a family of quasisymmetric functions called {\em Eulerian quasisymmetric functions}, which specialize to enumerators for the joint distribution of the permutation statistics, major index and excedance number on permutations of fixed cycle type. This family is analogous to a family of quasisymmetric functions that Gessel and Reutenauer used to study the joint distribution of major index and descent number on permutations of fixed cycle type. Our central result is a formula for the generating function for the Eulerian quasisymmetric functions, which specializes to a new and surprising $q$-analog of a classical formula of Euler for the exponential generating function of the Eulerian polynomials. This $q$-analog computes the joint distribution of excedance number and major index, the only of the four important Euler-Mahonian distributions that had not yet been computed. Our study of the Eulerian quasisymmetric functions also yields results that include the descent statistic and refine results of Gessel and Reutenauer. We also obtain $q$-analogs, $(q,p)$-analogs and quasisymmetric function analogs of classical results on the symmetry and unimodality of the Eulerian polynomials. Our Eulerian quasisymmetric functions refine symmetric functions that have occurred in various representation theoretic and enumerative contexts including MacMahon's study of multiset derangements, work of Procesi and Stanley on toric varieties of Coxeter complexes, Stanley's work on chromatic symmetric functions, and the work of the authors on the homology of a certain poset introduced by Björner and Welker.
It is shown that a refined version of a q-analogue of the Eulerian numbers together with the action, by conjugation, of the subgroup of the symmetric group $S_n$ generated by the $n$-cycle $(1,2,...,n)$ on the set of permutations of fixed cycle type and fixed number of excedances provides an instance of the cyclic sieving phenonmenon of Reiner, Stanton and White. The main tool is a class of symmetric functions recently introduced in work of two of the authors.
The notion of Rees product of posets was introduced by Björner and Welker, where they study connections between poset topology and commutative algebra. Björner and Welker conjectured and Jonsson proved that the dimension of the top homology of the Rees product of the truncated Boolean algebra $B_n \setminus \{0\}$ and the $n$-chain $C_n$ is equal to the number of derangements in the symmetric group $\mathfrak S_n$. Here we prove a refinement of this result, which involves the Eulerian numbers, and a $q$-analog of both the refinement and the original conjecture, which comes from replacing the Boolean algebra by the lattice of subspaces of the $n$-dimensional vector space over the $q$ element field, and involves the $(\maj,\exc)$-$q$-Eulerian polynomials studied in previous papers of the authors. Equivariant versions of the refinement and the original conjecture are also proved, as are type BC versions (in the sense of Coxeter groups) of the original conjecture and its $q$-analog.
We investigate the representation of a symmetric group $S_n$ on the homology of its Quillen complex at a prime $p$. For homology groups in small codimension, we derive an explicit formula for this representation in terms of the representations of symmetric groups on homology groups of $p$-uniform hypergraph matching complexes. We conjecture an explicit formula for the representation of $S_n$ on the top homology group of the corresponding hypergraph matching complex when $n \equiv 1 \bmod p$. Our conjecture follows from work of Bouc when $p=2$, and we prove the conjecture when $p=3$.
We introduce a family of quasisymmetric functions called {\em Eulerian quasisymmetric functions}, which have the property of specializing to enumerators for the joint distribution of the permutation statistics, major index and excedance number on permutations of fixed cycle type. This family is analogous to a family of quasisymmetric functions that Gessel and Reutenauer used to study the joint distribution of major index and descent number on permutations of fixed cycle type. Our central result is a formula for the generating function for the Eulerian quasisymmetric functions, which specializes to a new and surprising $q$-analog of a classical formula for the exponential generating function of the Eulerian polynomials. This $q$-analog computes the joint distribution of excedance number and major index, the only of the four important Euler-Mahonian distributions that had not yet been computed. Our study of the Eulerian quasisymmetric functions also yields results that include the descent statistic and refine results of Gessel and Reutenauer. We also obtain $q$-analogs, $(q,p)$-analogs and quasisymmetric function analogs of classical results on the symmetry and unimodality of the Eulerian polynomials. Our Eulerian quasisymmetric functions refine symmetric functions that have occurred in various representation theoretic and enumerative contexts such as in MacMahon's study of multiset derangements, in work of Procesi and Stanley on toric varieties of Coxeter complexes and in Stanley's work on symmetric chromatic polynomials. Here we present yet another occurence in connection with the homology of a poset introduced by Björner and Welker.
We prove the conjecture of A. Postnikov that (A) the number of regions in the inversion hyperplane arrangement associated with a permutation $w\in \Sn$ is at most the number of elements below $w$ in the Bruhat order, and (B) that equality holds if and only if $w$ avoids the patterns 4231, 35142, 42513 and 351624. Furthermore, assertion (A) is extended to all finite reflection groups. A byproduct of this result and its proof is a set of inequalities relating Betti numbers of complexified inversion arrangements to Betti numbers of closed Schubert cells. Another consequence is a simple combinatorial interpretation of the chromatic polynomial of the inversion graph of a permutation which avoids the above patterns.
In this research announcement we present a new q-analog of a classical formula for the exponential generating function of the Eulerian polynomials. The Eulerian polynomials enumerate permutations according to their number of descents or their number of excedances. Our q-Eulerian polynomials are the enumerators for the joint distribution of the excedance statistic and the major index. There is a vast literature on q-Eulerian polynomials which involve other combinations of Mahonian and Eulerian permutation statistics, but the combination of major index and excedance number seems to have been completely overlooked until now. We use symmetric function theory to prove our formula. In particular, we prove a symmetric function version of our formula, which involves an intriguing new class of symmetric functions. We also present connections with representations of the symmetric group on the homology of a poset recently introduced by Björner and Welker and on the cohomology of the toric variety associated with the Coxeter complex of the symmetric group, studied by Procesi, Stanley, Stembridge, Dolgachev and Lunts.
For positive integers k,n, we investigate the simplicial complex NM_k(n) of all graphs G on vertex set [n] such that every matching in G has size less than k. This complex (along with other associated cell complexes) is found to be homotopy equivalent to a wedge of spheres. The number and dimension of the spheres in the wedge are determined, and (partially conjectural) links to other combinatorially defined complexes are described. In addition we study for positive integers r,s and k the simplicial complex BNM_k(n) of all bipartite graphs G on bipartition [r] \cup [\bar{s}] such that there is no matching of size k in G, and obtain results similar to those obtained for NM_k(n).
Topological properties of the matching complex were first studied by Bouc in connection with Quillen complexes, and topological properties of the chessboard complex were first studied by Garst in connection with Tits coset complexes. Björner, Lovász, Vrécica and {Ž}ivaljević established bounds on the connectivity of these complexes and conjectured that these bounds are sharp. In this paper we show that the conjecture is true by establishing the nonvanishing of integral homology in the degrees given by these bounds. Moreover, we show that for sufficiently large $n$, the bottom nonvanishing homology of the matching complex $M_n$ is an elementary 3-group, improving a result of Bouc, and that the bottom nonvanishing homology of the chessboard complex $M_{n,n}$ is a 3-group of exponent at most 9. When $n \equiv 2 \bmod 3$, the bottom nonvanishing homology of $M_{n,n}$ is shown to be $\Z_3$. Our proofs rely on computer calculations, long exact sequences, representation theory, and tableau combinatorics.
In this paper, we first give formulas for the order polynomial $Ω(\Pw; t)$ and the Eulerian polynomial $e(\Pw; λ)$ of a finite labeled poset $(P, ω)$ using the adjacency matrix of what we call the $ω$-graph of $(P, ω)$. We then derive various recursion formulas for $Ω(\Pw; t)$ and $e(\Pw; λ)$ and discuss some applications of these formulas to Bernoulli numbers and Bernoulli polynomials. Finally, we give a recursive algorithm using a single linear operator on a vector space. This algorithm provides a uniform method to construct a family of new invariants for labeled posets $(\Pw)$, which includes the order polynomial $Ω(\Pw; t)$ and the invariant $\tilde e(\Pw; λ) =\frac {e(\Pw; λ)}{(1-λ)^{|P|+1}}$. The well-known quasi-symmetric function invariant of labeled posets and a further generalization of our construction are also discussed.
Complexes of (not) connected graphs, hypergraphs and their homology appear in the construction of knot invariants given by V. Vassiliev. In this paper we study the complexes of not $i$-connected $k$-hypergraphs on $n$ vertices. We show that the complex of not $2$-connected graphs has the homotopy type of a wedge of $(n-2)!$ spheres of dimension $2n-5$. This answers one of the questions raised by Vassiliev in connection with knot invariants. For this case the $S_n$-action on the homology of the complex is also determined. For complexes of not $2$-connected $k$-hypergraphs we provide a formula for the generating function of the Euler characteristic, and we introduce certain lattices of graphs that encode their topology. We also present partial results for some other cases. In particular, we show that the complex of not $(n-2)$-connected graphs is Alexander dual to the complex of partial matchings of the complete graph. For not $(n-3)$-connected graphs we provide a formula for the generating function of the Euler characteristic.