Descent c-Wilf Equivalence

Let $S_n$ denote the symmetric group. For any $\sigma \in S_n$, we let $\mathrm{des}(\sigma)$ denote the number of descents of $\sigma$, $\mathrm{inv}(\sigma)$ denote the number of inversions of $\sigma$, and $\mathrm{LRmin}(\sigma)$ denote the number of left-to-right minima of $\sigma$. For any sequence of statistics $\mathrm{stat}_1, \ldots \mathrm{stat}_k$ on permutations, we say two permutations $\alpha$ and $\beta$ in $S_j$ are $(\mathrm{stat}_1, \ldots \mathrm{stat}_k)$-c-Wilf equivalent if the generating function of $\prod_{i=1}^k x_i^{\mathrm{stat}_i}$ over all permutations which have no consecutive occurrences of $\alpha$ equals the generating function of $\prod_{i=1}^k x_i^{\mathrm{stat}_i}$ over all permutations which have no consecutive occurrences of $\beta$. We give many examples of pairs of permutations $\alpha$ and $\beta$ in $S_j$ which are $\mathrm{des}$-c-Wilf equivalent, $(\mathrm{des},\mathrm{inv})$-c-Wilf equivalent, and $(\mathrm{des},\mathrm{inv},\mathrm{LRmin})$-c-Wilf equivalent. For example, we will show that if $\alpha$ and $\beta$ are minimally overlapping permutations in $S_j$ which start with 1 and end with the same element and $\mathrm{des}(\alpha) = \mathrm{des}(\beta)$ and $\mathrm{inv}(\alpha) = \mathrm{inv}(\beta)$, then $\alpha$ and $\beta$ are $(\mathrm{des},\mathrm{inv})$-c-Wilf equivalent.Comment: arXiv admin note: text overlap with arXiv:1510.0431

Pattern Avoidance in Task-Precedence Posets

We have extended classical pattern avoidance to a new structure: multiple task-precedence posets whose Hasse diagrams have three levels, which we will call diamonds. The vertices of each diamond are assigned labels which are compatible with the poset. A corresponding permutation is formed by reading these labels by increasing levels, and then from left to right. We used Sage to form enumerative conjectures for the associated permutations avoiding collections of patterns of length three, which we then proved. We have discovered a bijection between diamonds avoiding 132 and certain generalized Dyck paths. We have also found the generating function for descents, and therefore the number of avoiders, in these permutations for the majority of collections of patterns of length three. An interesting application of this work (and the motivating example) can be found when task-precedence posets represent warehouse package fulfillment by robots, in which case avoidance of both 231 and 321 ensures we never stack two heavier packages on top of a lighter package.Comment: 17 page

Lattice Paths and Pattern-Avoiding Uniquely Sorted Permutations

Defant, Engen, and Miller defined a permutation to be uniquely sorted if it has exactly one preimage under West's stack-sorting map. We enumerate classes of uniquely sorted permutations that avoid a pattern of length three and a pattern of length four by establishing bijections between these classes and various lattice paths. This allows us to prove nine conjectures of Defant.Comment: 18 pages, 16 figures, new version with updated abstract and reference