On pp-adic valuations of colored pp-ary partitions

Let $m\in\N_{\geq 2}$ and for given $k\in\N_{+}$ consider the sequence $(A_{m,k}(n))_{n\in\N}$ defined by the power series expansion $$\prod_{n=0}^{\infty}\frac{1}{\left(1-x^{m^{n}}\right)^{k}}=\sum_{n=0}^{\infty}A_{m,k}(n)x^{n}.$$ The number $A_{m,k}(n)$ counts the number of representations of $n$ as sums of powers of $m$, where each summand has one among $k$ colors. In this note we prove that for each $p\in\mathbb{P}_{\geq 3}$ and $s\in\N_{+}$, the $p$-adic valuation of the number $A_{p,(p-1)(p^s-1)}(n)$ is equal to 1 for $n\geq p^s$. We also obtain some results concerning the behaviour of the sequence $(\nu_{p}(A_{p,(p-1)(up^s-1)}(n)))_{n\in\N}$ for fixed $u\in\{2,\ldots,p-1\}$ and $p\geq 3$. Our results generalize the earlier findings obtained for $p=2$ by Gawron, Miska and the first author.Comment: 10 pages, to appear in Monatshefte f\"{u}r Mathemati

The stability of the roommate problem revisited

The lack of stability in some matching problems suggests that alternative solution concepts to the core might be a step towards furthering our understanding of matching market performance. We propose absorbing sets as a solution for the class of roommate problems with strict preferences. This solution, which always exists, either gives the matchings in the core or predicts other matchings when the core is empty. Furthermore, it satisfies the interesting property of outer stability. We also determine the matchings in absorbing sets and find that in the case of multiple absorbing sets a similar structure is shared by all.roommate problem, core, absorbing sets

Heuristics for Longest Edge Selection in Simplicial Branch and Bound

Pre-print de la comunicacion presentada al ICCSA2015Simplicial partitions are suitable to divide a bounded area in branch and bound. In the iterative re nement process, a popular strategy is to divide simplices by their longest edge, thus avoiding needle-shaped simplices. A range of possibilities arises in higher dimensions where the number of longest edges in a simplex is greater than one. The behaviour of the search and the resulting binary search tree depend on the se- lected longest edge. In this work, we investigate different rules to select a longest edge and study the resulting efficiency of the branch and bound algorithm.Universidad de Málaga. Campus de Excelencia Internacional Andalucía Tech

Promotion and Rowmotion

We present an equivariant bijection between two actions--promotion and rowmotion--on order ideals in certain posets. This bijection simultaneously generalizes a result of R. Stanley concerning promotion on the linear extensions of two disjoint chains and recent work of D. Armstrong, C. Stump, and H. Thomas on root posets and noncrossing partitions. We apply this bijection to several classes of posets, obtaining equivariant bijections to various known objects under rotation. We extend the same idea to give an equivariant bijection between alternating sign matrices under rowmotion and under B. Wieland's gyration. Finally, we define two actions with related orders on alternating sign matrices and totally symmetric self-complementary plane partitions.Comment: 25 pages, 22 figures; final versio