A major index for matchings and set partitions

We introduce a statistic \pmaj on partitions of $[n]=\{1,2,..., n\}$, and show that it is equidistributed with the number of 2-crossings over partitions of $[n]$ with given sets of minimal block elements and maximal block elements. This generalizes the classical result of equidistribution for the permutation statistics inversion number and major index.Comment: 17 pages, 9 figure

Research on Combinatorial Statistics: Crossings and Nestings in Discrete Structures

We study the distribution of combinatorial statistics that exhibit a structure of crossings and nesting in various discrete structures, in particular, in set partitions, matchings, and fillings of moon polyominoes with entries 0 and 1. Let pi and y be two set partitions with the same number of blocks. Assume pi is a partition of [n]. For any integers l, m >̲ 0, let T (pi, l) be the set of partitions of [n + l] whose restrictions to the last n elements are isomorphic to pi, and T (pi, l, m) the subset of T (pi, l) consisting of those partitions with exactly m blocks. Similarly define T (pi, l) and T (y, l, m). We prove that if the statistic cr (ne), the number of crossings (nestings) of two edges, coincides on the sets T (pi, l) and T (pi, l) for l = 0; 1, then it coincides on T (pi, l, m) and T (y, l, m) for all l, m >̲ 0. These results extend the ones obtained by Klazar on the distribution of crossings and nestings for matchings. Moreover, we give a bijection between partially directed paths in the symmetric wedge y = +̲ x and matchings, which sends north steps to nestings. This gives a bijective proof of a result of E. J. Janse van Rensburg, T. Prellberg, and A. Rechnitzer that was first discovered through the corresponding generating functions: the number of partially directed paths starting at the origin confined to the symmetric wedge y = +̲ x with k north steps is equal to the number of matchings on [2n] with k nestings. Furthermore, we propose a major index statistic on 01-fillings of moon polyominoes which, when specialized to certain shapes, reduces to the major index for permutations and set partitions. We consider the set F(M, s, A) of all 01-fillings of a moon polyomino M with given column sum s whose empty rows are A, and prove that this major index has the same distribution as the number of north-east chains, which are the natural extension of inversions (resp. crossings) for permutations (resp. set partitions). Hence our result generalizes the classical equidistribution results for the permutation statistics inv and maj. Two proofs are presented. The first is an algebraic one using generating functions, and the second is a bijection on 01-fillings of moon polyominoes in the spirit of Foata's second fundamental transformation on words and permutations

Mixed Statistics on 01-Fillings of Moon Polyominoes

We establish a stronger symmetry between the numbers of northeast and southeast chains in the context of 01-fillings of moon polyominoes. Let \M be a moon polyomino with $n$ rows and $m$ columns. Consider all the 01-fillings of \M in which every row has at most one 1. We introduce four mixed statistics with respect to a bipartition of rows or columns of \M. More precisely, let $S \subseteq \{1,2,..., n\}$ and $\mathcal{R}(S)$ be the union of rows whose indices are in $S$. For any filling $M$, the top-mixed (resp. bottom-mixed) statistic $\alpha(S; M)$ (resp. $\beta(S; M)$) is the sum of the number of northeast chains whose top (resp. bottom) cell is in $\mathcal{R}(S)$, together with the number of southeast chains whose top (resp. bottom) cell is in the complement of $\mathcal{R}(S)$. Similarly, we define the left-mixed and right-mixed statistics $\gamma(T; M)$ and $\delta(T; M)$, where $T$ is a subset of the column index set $\{1,2,..., m\}$. Let $\lambda(A; M)$ be any of these four statistics $\alpha(S; M)$, $\beta(S; M)$, $\gamma(T; M)$ and $\delta(T; M)$, we show that the joint distribution of the pair $(\lambda(A; M), \lambda(\bar A; M))$ is symmetric and independent of the subsets $S, T$. In particular, the pair of statistics $(\lambda(A;M), \lambda(\bar A; M))$ is equidistributed with (\se(M),\ne(M)), where \se(M) and $\ne(M)$ are the numbers of southeast chains and northeast chains of $M$, respectively.Comment: 20 pages, 6 figure

Avoidance of Partitions of a Three-element Set

Klazar defined and studied a notion of pattern avoidance for set partitions, which is an analogue of pattern avoidance for permutations. Sagan considered partitions which avoid a single partition of three elements. We enumerate partitions which avoid any family of partitions of a 3-element set as was done by Simion and Schmidt for permutations. We also consider even and odd set partitions. We provide enumerative results for set partitions restricted by generalized set partition patterns, which are an analogue of the generalized permutation patterns of Babson and Steingr{\'{\i}}msson. Finally, in the spirit of work done by Babson and Steingr{'{\i}}msson, we will show how these generalized partition patterns can be used to describe set partition statistics.Comment: 23 pages, 2 tables, 1 figure, to appear in Advances in Applied Mathematic

Grafalgo - A Library of Graph Algorithms and Supporting Data Structures (revised)

This report provides an (updated) overview of {\sl Grafalgo}, an open-source library of graph algorithms and the data structures used to implement them. The programs in this library were originally written to support a graduate class in advanced data structures and algorithms at Washington University. Because the code's primary purpose was pedagogical, it was written to be as straightforward as possible, while still being highly efficient. Grafalgo is implemented in C++ and incorporates some features of C++11. The library is available on an open-source basis and may be downloaded from https://code.google.com/p/grafalgo/. Source code documentation is at www.arl.wustl.edu/\textasciitilde jst/doc/grafalgo. While not designed as production code, the library is suitable for use in larger systems, so long as its limitations are understood. The readability of the code also makes it relatively straightforward to extend it for other purposes

Approximating Holant problems by winding

We give an FPRAS for Holant problems with parity constraints and not-all-equal constraints, a generalisation of the problem of counting sink-free-orientations. The approach combines a sampler for near-assignments of "windable" functions -- using the cycle-unwinding canonical paths technique of Jerrum and Sinclair -- with a bound on the weight of near-assignments. The proof generalises to a larger class of Holant problems; we characterise this class and show that it cannot be extended by expressibility reductions. We then ask whether windability is equivalent to expressibility by matchings circuits (an analogue of matchgates), and give a positive answer for functions of arity three