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Register a new userPlane permutations and applications to a result of Zagier-Stanley and distances of permutations
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In this paper, we introduce plane permutations, i.e. pairs $mathfrak{p}=(s,pi)$ where $s$ is an $n$-cycle and $pi$ is an arbitrary permutation, represented as a two-row array. Accordingly a plane permutation gives rise to three distinct permutations: the permutation induced by the upper horizontal ($s$), the vertical $pi$) and the diagonal ($D_{mathfrak{p}}$) of the array. The latter can also be viewed as the three permutations of a hypermap. In particular, a map corresponds to a plane permutation, in which the diagonal is a fixed point-free involution. We study the transposition action on plane permutations obtained by permuting their diagonal-blocks. We establish basic properties of plane permutations and study transpositions and exceedances and derive various enumerative results. In particular, we prove a recurrence for the number of plane permutations having a fixed diagonal and $k$ cycles in the vertical, generalizing Chapuys recursion for maps filtered by the genus. As applications of this framework, we present a combinatorial proof of a result of Zagier and Stanley, on the number of $n$-cycles $omega$, for which the product $omega(1~2~cdots ~n)$ has exactly $k$ cycles. Furthermore, we integrate studies on the transposition and block-interchange distance of permutations as well as the reversal distance of signed permutations. Plane permutations allow us to generalize and recover various lower bounds for transposition and block-interchange distances and to connect reversals with block-interchanges.
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In this paper, we enumerate the pairs of permutations that are long cycles and whose product has a given cycle-type. Our main result is a simple relation concerning the desired numbers for a few related cycle-types. The relation refines a formula of the number of pairs of long cycles whose product has $k$ cycles independently obtained by Zagier and Stanley relying on group characters, and was previously obtained by F{e}ray and Vassilieva by counting some colored permutations first and then relying on some algebraic computations in the ring of symmetric functions. Our approach here is simpler and combinatorial.
In this paper we generalize permutations to plane permutations. We employ this framework to derive a combinatorial proof of a result of Zagier and Stanley, that enumerates the number of $n$-cycles $omega$, for which $omega(12cdots n)$ has exactly $k$ cycles. This quantity is $0$, if $n-k$ is odd and $frac{2C(n+1,k)}{n(n+1)}$, otherwise, where $C(n,k)$ is the unsigned Stirling number of the first kind. The proof is facilitated by a natural transposition action on plane permutations which gives rise to various recurrences. Furthermore we study several distance problems of permutations. It turns out that plane permutations allow to study transposition and block-interchange distance of permutations as well as the reversal distance of signed permutations. Novel connections between these different distance problems are established via plane permutations.
In this paper, we first obtain some analogues of a formula of Zagier (1995) and Stanley (2011). For instance, we prove that the number of pairs of $n$-cycles whose product has $k$ cycles and has $m$ given elements contained in distinct cycles (or separated) is given by $$ frac{2 (n-1)! C_m(n+1,k)}{(n+m)(n+1-m)} $$ when $n-k$ is even, where $C_m(n,k)$ is the number of permutations of $n$ elements having $k$ cycles and separating $m$ given elements. As consequences, we obtain the formulas for certain separation probabilities due to Du and Stanley, answering a call of Stanley for simple combinatorial proofs. Furthermore, we obtain the expectation and variance of the number of fixed points in the product of two random $n$-cycles.
We extend an observation due to Stong that the distribution of the number of degree $d$ irreducible factors of the characteristic polynomial of a random $n times n$ matrix over a finite field $mathbb{F}_{q}$ converges to the distribution of the number of length $d$ cycles of a random permutation in $S_{n}$, as $q rightarrow infty$, by having any finitely many choices of $d$, say $d_{1}, dots, d_{r}$. This generalized convergence will be used for the following two applications: the distribution of the cokernel of an $n times n$ Haar-random $mathbb{Z}_{p}$-matrix when $p rightarrow infty$ and a matrix version of Landaus theorem that estimates the number of irreducible factors of a random characteristic polynomial for large $n$ when $q rightarrow infty$.
This paper develops an analogy between the cycle structure of, on the one hand, random permutations with cycle lengths restricted to lie in an infinite set $S$ with asymptotic density $sigma$ and, on the other hand, permutations selected according to the Ewens distribution with parameter $sigma$. In particular we show that the asymptotic expected number of cycles of random permutations of $[n]$ with all cycles even, with all cycles odd, and chosen from the Ewens distribution with parameter 1/2 are all ${1 over 2} log n + O(1)$, and the variance is of the same order. Furthermore, we show that in permutations of $[n]$ chosen from the Ewens distribution with parameter $sigma$, the probability of a random element being in a cycle longer than $gamma n$ approaches $(1-gamma)^sigma$ for large $n$. The same limit law holds for permutations with cycles carrying multiplicative weights with average $sigma$. We draw parallels between the Ewens distribution and the asymptotic-density case and explain why these parallels should exist using permutations drawn from weighted Boltzmann distributions.
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