Stirling permutation

{{Short description|Type of permutation in combinatorial mathematics}}

In combinatorial mathematics, a Stirling permutation of order k is a permutation of the multiset 1, 1, 2, 2, ..., k, k (with two copies of each value from 1 to k) with the additional property that, for each value i appearing in the permutation, any values between the two copies of i are larger than i. For instance, the 15 Stirling permutations of order three are

:1,1,2,2,3,3;   1,2,2,1,3,3;   2,2,1,1,3,3;

:1,1,2,3,3,2;   1,2,2,3,3,1;   2,2,1,3,3,1;

:1,1,3,3,2,2;   1,2,3,3,2,1;   2,2,3,3,1,1;

:1,3,3,1,2,2;   1,3,3,2,2,1;   2,3,3,2,1,1;

:3,3,1,1,2,2;   3,3,1,2,2,1;   3,3,2,2,1,1.

The number of Stirling permutations of order k is given by the double factorial (2k − 1)!!. Stirling permutations were introduced by {{harvtxt|Gessel|Stanley|1978}} in order to show that certain numbers (the numbers of Stirling permutations with a fixed number of descents) are non-negative. They chose the name because of a connection to certain polynomials defined from the Stirling numbers, which are in turn named after 18th-century Scottish mathematician James Stirling.{{citation

| last1 = Gessel | first1 = Ira | author1-link = Ira Gessel

| last2 = Stanley | first2 = Richard P. | author2-link = Richard P. Stanley

| doi = 10.1016/0097-3165(78)90042-0

| issue = 1

| journal = Journal of Combinatorial Theory

| mr = 0462961

| pages = 24–33

| series = Series A

| title = Stirling polynomials

| volume = 24

| year = 1978| doi-access = free

}}.

File:Stirling permutation Euler tour.svg of a plane tree with its edges labeled by construction order]]

Stirling permutations may be used to describe the sequences by which it is possible to construct a rooted plane tree with k edges by adding leaves one by one to the tree. For, if the edges are numbered by the order in which they were inserted, then the sequence of numbers in an Euler tour of the tree (formed by doubling the edges of the tree and traversing the children of each node in left to right order) is a Stirling permutation. Conversely every Stirling permutation describes a tree construction sequence, in which the next edge closer to the root from an edge labeled i is the one whose pair of values most closely surrounds the pair of i values in the permutation.{{citation

| last = Janson | first = Svante | authorlink = Svante Janson

| arxiv = 0803.1129

| contribution = Plane recursive trees, Stirling permutations and an urn model

| mr = 2508813

| pages = 541–547

| publisher = Assoc. Discrete Math. Theor. Comput. Sci., Nancy

| series = Discrete Math. Theor. Comput. Sci. Proc., AI

| title = Fifth Colloquium on Mathematics and Computer Science

| year = 2008| bibcode = 2008arXiv0803.1129J}}.

Stirling permutations have been generalized to the permutations of a multiset with more than two copies of each value.{{citation

| last1 = Klingsberg | first1 = Paul

| last2 = Schmalzried | first2 = Cynthia

| contribution = A family of constructive bijections involving Stirling permutations

| series = Congressus Numerantium

| mr = 1140465

| pages = 11–15

| title = Proceedings of the Twenty-First Southeastern Conference on Combinatorics, Graph Theory, and Computing (Boca Raton, Florida, 1990)

| volume = 78

| year = 1990}}. Researchers have also studied the number of Stirling permutations that avoid certain patterns.{{citation

| last1 = Kuba | first1 = Markus

| last2 = Panholzer | first2 = Alois

| doi = 10.1016/j.disc.2012.07.011

| issue = 21

| journal = Discrete Mathematics

| mr = 2957938

| pages = 3179–3194

| title = Enumeration formulæ for pattern restricted Stirling permutations

| volume = 312

| year = 2012| doi-access = free

}}.