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On the number of cycles of given length of a free word in several random permutations
Author(s) -
Nica Alexandru
Publication year - 1994
Publication title -
random structures and algorithms
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.314
H-Index - 69
eISSN - 1098-2418
pISSN - 1042-9832
DOI - 10.1002/rsa.3240050506
Subject(s) - combinatorics , mathematics , permutation (music) , word (group theory) , word length , random permutation , poisson distribution , limit (mathematics) , group (periodic table) , type (biology) , discrete mathematics , symmetric group , physics , statistics , geometry , mathematical analysis , computer science , quantum mechanics , ecology , natural language processing , acoustics , biology
Let w ≠ 1 be a free word in the symbols g 1 ,…, g k and their inverses (i.e., an element of the free group F k ). For any s 1 ,…, s k , in the group s n of all permutation of n objects, we denote by w ( s 1 ,…, s k ) ϵ S n the permutation obtained by replacing g 1 ,…, g k with s 1 ,…, s k in the expression of w . Let X w,L ( n )( s 1 ,…, s k ) denote the number of cycles of length L of w ( s 1 ,…, s k ). For fixed w and L , we show that X w,L ( n ) , viewed as a random variable on S n k , has (for n →∞) a Poisson‐type limit distribution, which can be computed precisely. © 1994 John Wiley & Sons, Inc.
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