Two enumerative results on cycles of permutations
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Stanley_Two Enumerative.pdf
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114.65 KB
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Author(s)
Stanley, Richard P.
Date Issued
February 2011
Journal
European Journal of Combinatorics
Publisher
Elsevier
Citation
Stanley, Richard P. “Two Enumerative Results on Cycles of Permutations.” European Journal of Combinatorics 32, no. 6 (August 2011): 937–943.
Version
Author's final manuscript
Abstract
Answering a question of Bona, it is shown that for n≥2 the probability that 1 and 2 are in the same cycle of a product of two n-cycles on the set {1,2,…,n} is 1/2 if n is odd and 1/2 - 2/(n-1)(n+2) if n is even. Another result concerns the polynomial P[subscript λ](q) = ∑[subscript w]q[superscript κ]((1,2,…,n)⋅w), where w ranges over all permutations in the symmetric group S[subscript n] of cycle type λ, (1,2,…,n) denotes the n-cycle 1→2→⋯→n→1, and κ(v) denotes the number of cycles of the permutation v. A formula is obtained for P[subscript λ](q) from which it is deduced that all zeros of P[subscript λ](q) have real part 0.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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Creative Commons Attribution-Noncommercial-NoDerivatives
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DOI of Published Version
https://doi.org/10.1016/j.ejc.2011.01.011
