Stirling permutation

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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, the values between the two copies of i are larger than i. Stirling permutation_sentence_0

For instance, the 15 Stirling permutations of order three are Stirling permutation_sentence_1

Stirling permutation_description_list_0

  • 1,1,2,2,3,3;   1,2,2,1,3,3;   2,2,1,1,3,3;Stirling permutation_item_0_0
  • 1,1,2,3,3,2;   1,2,2,3,3,1;   2,2,1,3,3,1;Stirling permutation_item_0_1
  • 1,1,3,3,2,2;   1,2,3,3,2,1;   2,2,3,3,1,1;Stirling permutation_item_0_2
  • 1,3,3,1,2,2;   1,3,3,2,2,1;   2,3,3,2,1,1;Stirling permutation_item_0_3
  • 3,3,1,1,2,2;   3,3,1,2,2,1;   3,3,2,2,1,1.Stirling permutation_item_0_4

The number of Stirling permutations of order k is given by the double factorial (2k − 1)!!. Stirling permutation_sentence_2

Stirling permutations were introduced by in order to show that certain numbers (the numbers of Stirling permutations with a fixed number of descents) are non-negative. Stirling permutation_sentence_3

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. Stirling permutation_sentence_4

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. Stirling permutation_sentence_5

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. Stirling permutation_sentence_6

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. Stirling permutation_sentence_7

Stirling permutations have been generalized to the permutations of a multiset with more than two copies of each value. Stirling permutation_sentence_8

Researchers have also studied the number of Stirling permutations that avoid certain patterns. Stirling permutation_sentence_9

See also Stirling permutation_section_0

Stirling permutation_unordered_list_1

  • Langford pairing, a different type of permutation of the same multisetStirling permutation_item_1_5

Credits to the contents of this page go to the authors of the corresponding Wikipedia page: permutation.