A permutation a1a2 ···an of 1,2,...,n is alternating if a1 > a2 a4 < ···. We survey some aspects of the theory of alternating permu- tations, beginning with the famous result of Andre that if En is the number of alternating permutations of 1,2,...,n, then P n�0 En x n n! = secx + tanx. Topics include refinements and q-analogues of En, various occurrences of En in mathematics, longest alternating subsequences of permutations, umbral enu- meration of special classes of alternating permutations, and the connection between alternating permutations and the cd-index of the symmetric group. Dedicated to Reza Khosrovshahi on the occasion of his 70th birthday

/pdf/a-survey-of-alternating-permutations-264ijjsqru.pdf

A Survey of Alternating Permutations

On the descent numbers and major indices for the hyperoctahedral group

Generalized Stirling permutations, families of increasing trees and urn models

Stable multivariate Eulerian polynomials and generalized Stirling permutations

P -partitions and q -Stirling numbers

The r -multipermutations

Enumeration of generalized lattice paths by string types, peaks, and ascents

We study generalized small Schroder paths in the sense of arbitrary sizes of steps. A generalized small Schroder path is a generalized lattice path from $(0,0)$ to $(2n,0)$ with the step set of  $\{(k,k), (l,-l), (2r,0)\, |\, k,l,r \in {\bf P}\}$, where ${\bf P}$ is the set of positive integers, which never goes below the $x$-axis, and with no horizontal steps at level 0.  We find a bijection between 5-colored Dyck paths and generalized small Schroder paths, proving that the number of generalized small Schroder paths is equal to $\sum_{k=1}^{n} N(n,k)5^{n-k}$ for $n\geq 1$.

/pdf/generalized-small-schroder-numbers-3jayyvy2xj.pdf

SeungKyung Park

Papers

The r -multipermutations

P -partitions and q -Stirling numbers

Enumeration of generalized lattice paths by string types, peaks, and ascents

Generalized Small Schroder numbers

The maximal-inversion statistic and pattern-avoiding permutations