Geometrically constructed bases for homology of partition lattices of types A, B and D

We use the theory of hyperplane arrangements to construct natural bases for the homology of partition lattices of types A, B and D. This extends and explains the "splitting basis" for the homology of the partition lattice given in [Wa96], thus answering a question asked by R. Stanley. More explicitly, the following general technique is presented and utilized. Let A be a central and essential hyperplane arrangement in R^d. Let R_1,...,R_k be the bounded regions of a generic hyperplane section of A. We show that there are induced polytopal cycles \rho_{R_i} in the homology of the proper part \bar{L_A} of the intersection lattice such that {\rho_{R_i}}_{i=1,...,k} is a basis for \tilde H_{d-2}(\bar{L_A}). This geometric method for constructing combinatorial homology bases is applied to the Coxeter arrangements of types A, B and D, and to some interpolating arrangements.Comment: 29 pages, 4 figure

Eulerian quasisymmetric functions

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 Bj\"orner and Welker.Comment: Final version; to appear in Advances in Mathematics; 52 pages; this paper was originally part of the longer paper arXiv:0805.2416v1, which has been split into three paper

Minimum cost partitions of a rectangle

We consider the problem of partitioning (in a certain manner) a rectangle into n regions of equal area so that the total lengths of the boundaries is a minimum. A closed form solution to the problem is presented

σ-Restricted growth functions and p,q-stirling numbers

AbstractThe restricted growth functions are known to encode set partitions. They are words whose subword of leftmost occurrences is the identity permutation. We generalize the notion of restricted growth function by considering words whose subword of leftmost occurrences is a fixed general permutation. We prove a natural generalization of results of Wachs and White which state that the enumerators for the joint distribution of two pairs of inversion like statistics on restricted growth functions are the p, q-Stirling numbers