Two enumerative results on cycles of permutations
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...
| প্রধান লেখক: | |
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| অন্যান্য লেখক: | |
| বিন্যাস: | প্রবন্ধ |
| ভাষা: | en_US |
| প্রকাশিত: |
Elsevier
2015
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| অনলাইন ব্যবহার করুন: | http://hdl.handle.net/1721.1/98849 https://orcid.org/0000-0003-3123-8241 |
| সংক্ষিপ্ত: | 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. |
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