Odd permutations are nicer than even ones
نویسندگان
چکیده
We give simple combinatorial proofs of some formulas for the number of factorizations of permutations in Sn as a product of two n-cycles, or of an n-cycle and an (n − 1)-cycle. © 2012 Elsevier Ltd. All rights reserved.
منابع مشابه
Combinatorics in the group of parity alternating permutations
We call a permutation parity alternating, if its entries assume even and odd integers alternately. Parity alternating permutations form a subgroup of the symmetric group. This paper deals with such permutations classified by two permutation statistics; the numbers of ascents and inversions. It turns out that they have a close relationship to signed Eulerian numbers. The approach is based on a s...
متن کاملAnalogues of Up-down Permutations for Colored Permutations
André proved that secx is the generating function of all up-down permutations of even length and tanx is the generating function of all up-down permutation of odd length. There are three equivalent ways to define up-down permutations in the symmetric group Sn. That is, a permutation σ in the symmetric group Sn is an updown permutation if either (i) the rise set of σ consists of all the odd numb...
متن کاملA Study of Eulerian Numbers for Permutations in the Alternating Group
The cardinalities of the sets of even and odd permutations with a given ascent number are investigated by an operator that was introduced by the author. We will deduce the recurrence relations for such Eulerian numbers of even and odd permutations, and deduce divisibility properties by prime powers concerning them and some related numbers.
متن کاملA note on the total number of cycles of even and odd permutations
We prove bijectively that the total number of cycles of all even permutations of [n] = {1, 2, . . . , n} and the total number of cycles of all odd permutations of [n] differ by (−1)(n − 2)!, which was proposed by Miklós Bóna. We also prove bijectively the following more general identity: n
متن کاملRefined sign-balance on 321-avoiding permutations
The number of even 321-avoiding permutations of length n is equal to the number of odd ones if n is even, and exceeds it by the n−1 2 th Catalan number otherwise. We present an involution that proves a refinement of this sign-balance property respecting the length of the longest increasing subsequence of the permutation. In addition, this yields a combinatorial proof of a recent analogous resul...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید
ثبت ناماگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید
ورودعنوان ژورنال:
- Eur. J. Comb.
دوره 33 شماره
صفحات -
تاریخ انتشار 2012