In combinatorial mathematics, a Stirling permutation of order ''k'' is a

permutation In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or p ...

of the

multiset In mathematics, a multiset (or bag, or mset) is a modification of the concept of a set that, unlike a set, allows for multiple instances for each of its elements. The number of instances given for each element is called the multiplicity of that ...

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''. 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 In mathematics, the double factorial or semifactorial of a number , denoted by , is the product of all the integers from 1 up to that have the same parity (odd or even) as . That is, :n!! = \prod_^ (n-2k) = n (n-2) (n-4) \cdots. For even , the ...

(2''k'' − 1)!!. 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. They chose the name because of a connection to certain

polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An ex ...

s defined from the Stirling numbers, which are in turn named after 18th-century Scottish mathematician James Stirling. Stirling permutation Euler tour

Stirling permutations may be used to describe the sequences by which it is possible to construct a rooted

plane tree ''Platanus'' is a genus consisting of a small number of tree species native to the Northern Hemisphere. They are the sole living members of the family Platanaceae. All mature members of ''Platanus'' are tall, reaching in height. All except ...

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. Stirling permutations have been generalized to the permutations of a multiset with more than two copies of each value. Researchers have also studied the number of Stirling permutations that avoid certain patterns..

References

{{reflist Permutations Combinatorics

See also

References