A congruence for the number of alternating permutations

We present a new proof of a result of Knuth and Buckholtz concerning the period of the number of alternating congruences modulo an odd prime. The proof is based on properties of special functions, specifically the polylogarithm, Dirichlet eta and beta functions, and Stirling numbers of the second kind.

Expansions of a Chord Diagram and Alternating Permutations

A chord diagram is a set of chords of a circle such that no pair of chords has a common endvertex. A chord diagram $E$ with $n$ chords is called an $n$-crossing if all chords of $E$ are mutually crossing. A chord diagram $E$ is called nonintersecting if $E$ contains no $2$-crossing. For a chord diagram $E$ having a $2$-crossing $S = \{ x_1 x_3, x_2 x_4 \}$, the expansion of $E$ with respect to $S$ is to replace $E$ with $E_1 = (E \setminus S) \cup \{ x_2 x_3, x_4 x_1 \}$ or $E_2 = (E \setminus S) \cup \{ x_1 x_2, x_3 x_4 \}$. It is shown that there is a one-to-one correspondence between the multiset of all nonintersecting chord diagrams generated from an $n$-crossing with a finite sequence of expansions and the set of alternating permutations of order $n+1$.

Ascent-Descent Young Diagrams and Pattern Avoidance in Alternating Permutations

We investigate pattern avoidance in alternating permutations and an alternating analogue of Young diagrams. In particular, using an extension of Babson and West's notion of shape-Wilf equivalence described in our recent paper (with N. Gowravaram), we generalize results of Backelin, West, and Xin and Ouchterlony to alternating permutations. Unlike Ouchterlony and Bóna's bijections, our bijections are not the restrictions of Backelin, West, and Xin's bijections to alternating permutations. This paper is the second of a two-paper series presenting the work of Beyond alternating permutations: Pattern avoidance in Young diagrams and tableaux (with N. Gowravaram, arXiv:1301.6796v1). The first paper in the series is Beyond alternating permutations: Pattern avoidance in Young diagrams and tableaux (with N. Gowravaram, Electronic Journal of Combinatorics 20(4):#P17, 2013).

Parity-alternating permutations and successions

The study of parity-alternating permutations of {1, 2, … n} is extended to permutations containing a prescribed number of parity successions — adjacent pairs of elements of the same parity. Several enumeration formulae are computed for permutations containing a given number of parity successions, in conjunction with further parity and length restrictions. The objects are classified using direct construction and elementary combinatorial techniques. Analogous results are derived for circular permutations.

Beyond Alternating Permutations: Pattern Avoidance in Young Diagrams and Tableaux

We investigate pattern avoidance in alternating permutations and generalizations thereof. First, we study pattern avoidance in an alternating analogue of Young diagrams. In particular, we extend Babson-West's notion of shape-Wilf equivalence to apply to alternating permutations and so generalize results of Backelin-West-Xin and Ouchterlony to alternating permutations. Second, we study pattern avoidance in the more general context of permutations with restricted ascents and descents. We consider a question of Lewis regarding permutations that are the reading words of thickened staircase Young tableaux, that is, permutations that have $k-1$ ascents followed by a descent, followed by $k-1$ ascents, et cetera. We determine the relative sizes of the sets of pattern-avoiding $(k-1)$-ascent permutations in terms of the forbidden pattern. Furthermore, inequalities in the sizes of sets of pattern-avoiding permutations in this context arise from further extensions of shape-equivalence type enumerations. This paper is the first of a two-paper series presenting the work of Beyond alternating permutations: Pattern avoidance in Young diagrams and tableaux (arXiv:1301.6796v1). The second in the series is Ascent-descent Young diagrams and pattern avoidance in alternating permutations (by the second author, submitted).

On Wilf Equivalence for Alternating Permutations

In this paper, we obtain several new classes of Wilf-equivalent patterns for alternating permutations. In particular, we prove that for any nonempty pattern $\tau$, the patterns $12\ldots k\oplus\tau$ and $k\ldots 21\oplus\tau$ are Wilf-equivalent for alternating permutations, paralleling a result of Backelin, West, and Xin for Wilf equivalence for permutations.