05937096

j

05937096 Stanley, Richard P. Two enumerative results on cycles of permutations. Eur. J. Comb. 32, No. 6, 937-943 (2011). 2011 Elsevier Science (Academic Press), London EN Zbl 1127.05001

doi:10.1016/j.ejc.2011.01.011

Summary: Answering a question of {\it M. B\'ona} [A walk through combinatorics. An introduction to enumeration and graph theory. 2nd ed. Hackensack, NJ: World Scientific. xviii, 469 p. (2006; Zbl 1127.05001)], it is shown that for $n\ge 2$ the probability that 1 and 2 are in the same cycle of a product of two $n$-cycles on the set $\{1,2,\dots,n\}$ is $1/2$ if $n$ is odd and ${1\over 2}- {2\over(n-1)(n+2)}$ if $n$ is even. Another result concerns the polynomial $$P_\lambda(q)= \sum_w q^{\kappa((1,2,\dots, n)\cdot w)},$$ where $w$ ranges over all permutations in the symmetric group ${\germ S}_n$ of cycle type $\lambda$, $(1,2,\dots,n)$ denotes the $n$-cycle $1\to 2\to\cdots\to n\to 1$, and $\kappa(v)$ denotes the number of cycles of the permutation $v$. A formula is obtained for $P_\lambda(q)$ from which it is deduced that all zeros of $P_\lambda(q)$ have real part 0.