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Generalizing the decomposition of a connected planar graph into a tree and a dual tree, we prove a combinatorial analog of the classic Helmholz-Hodge decomposition of a smooth vector field. Specifically, we show that for every polyhedral complex, $K$, and every dimension, $p$, there is a partition of the set of $p$-cells into a maximal $p$-tree, a maximal $p$-cotree, and a collection of $p$-cells whose cardinality is the $p$-th Betti number of $K$. Given an ordering of the $p$-cells, this tri-partition is unique, and it can be computed by a matrix reduction algorithm that also constructs canonical bases of cycle and boundary groups.
The symmetric group $mathfrak{S}_n$ acts on the polynomial ring $mathbb{Q}[mathbf{x}_n] = mathbb{Q}[x_1, dots, x_n]$ by variable permutation. The invariant ideal $I_n$ is the ideal generated by all $mathfrak{S}_n$-invariant polynomials with vanishing
We study statistics on ordered set partitions whose generating functions are related to $p,q$-Stirling numbers of the second kind. The main purpose of this paper is to provide bijective proofs of all the conjectures of stein (Arxiv:math.CO/0605670).
For each skew shape we define a nonhomogeneous symmetric function, generalizing a construction of Pak and Postnikov. In two special cases, we show that the coefficients of this function when expanded in the complete homogeneous basis are given in ter
In this work, we give combinatorial proofs for generating functions of two problems, i.e., flushed partitions and concave compositions of even length. We also give combinatorial interpretation of one problem posed by Sylvester involving flushed parti
We study two families of probability measures on integer partitions, which are Schur measures with parameters tuned in such a way that the edge fluctuations are characterized by a critical exponent different from the generic $1/3$. We find that the f