Pattern restricted quasi-Stirling permutations

We define a variation of Stirling permutations, called quasi-Stirling permutations, to be permutations on the multiset $\{1,1,2,2,\ldots, n,n\}$ that avoid the patterns 1212 and 2121. Their study is motivated by a known relationship between Stirling permutations and increasing ordered rooted labeled trees. We construct a bijection between quasi-Stirling permutations and the set of ordered rooted labeled trees and investigate pattern avoidance for these permutations

Descents on quasi-Stirling permutations

Stirling permutations were introduced by Gessel and Stanley, who used their enumeration by the number of descents to give a combinatorial interpretation of certain polynomials related to Stirling numbers. Quasi-Stirling permutations, which can be viewed as labeled noncrossing matchings, were introduced by Archer et al. as a natural extension of Stirling permutations. Janson's correspondence between Stirling permutations and labeled increasing plane trees extends to a bijection between quasi-Stirling permutations and the same set of trees without the increasing restriction. Archer et al. posed the problem of enumerating quasi-Stirling permutations by the number of descents, and conjectured that there are $(n+1)^{n-1}$ such permutations of size $n$ having the maximum number of descents. In this paper we prove their conjecture, and we give the generating function for quasi-Stirling permutations by the number of descents, expressed as a compositional inverse of the generating function of Eulerian polynomials. We also find the analogue for quasi-Stirling permutations of the main result from Gessel and Stanley's paper. We prove that the distribution of descents on these permutations is asymptotically normal, and that the roots of the corresponding quasi-Stirling polynomials are all real, in analogy to B\'ona's results for Stirling permutations. Finally, we generalize our results to a one-parameter family of permutations that extends $k$-Stirling permutations, and we refine them by also keeping track of the number of ascents and the number of plateaus

Signature Catalan Combinatorics

The Catalan numbers constitute one of the most important sequences in combinatorics. Catalan objects have been generalized in various directions, including the classical Fuss-Catalan objects and the rational Catalan generalization of Armstrong-Rhoades-Williams. We propose a wider generalization of these families indexed by a composition $s$ which is motivated by the combinatorics of planar rooted trees; when $s=(2,...,2)$ and $s=(k+1,...,k+1)$ we recover the classical Catalan and Fuss-Catalan combinatorics, respectively. Furthermore, to each pair $(a,b)$ of relatively prime numbers we can associate a signature that recovers the combinatorics of rational Catalan objects. We present explicit bijections between the resulting $s$-Catalan objects, and a fundamental recurrence that generalizes the fundamental recurrence of the classical Catalan numbers. Our framework allows us to define signature generalizations of parking functions which coincide with the generalized parking functions studied by Pitman-Stanley and Yan, as well as generalizations of permutations which coincide with the notion of Stirling multipermutations introduced by Gessel-Stanley. Some of our constructions differ from the ones of Armstrong-Rhoades-Williams, however as a byproduct of our extension, we obtain the additional notions of rational permutations and rational trees.Comment: 42 pages, 31 figure

Analysis of statistics for generalized Stirling permutations

In this work we give a study of generalizations of Stirling permutations, a restricted class of permutations of multisets introduced by Gessel and Stanley [15]. First we give several bijections between such generalized Stirling permutations and various families of increasing trees extending the known correspondences of [20, 21]. Then we consider several permutation statistics of interest for generalized Stirling permutations as the number of left-to-right minima, the number of left-to-right maxima, the number of blocks of specified sizes, the distance between occurrences of elements, and the number of inversions. For all these quantities we give a distributional study, where the established connections to increasing trees turn out be very useful. To obtain the exact and limiting distribution results we use several techniques ranging from generating functions, connections to urn models, martingales and Stein’s method