Cyclic Sieving Phenomenon on Matchings
نویسنده
چکیده
School of Mathematical and Statistical Sciences, Arizona State University Abstract. Reiner-Stanton-White defined the cyclic sieving phenomenon (CSP) associated to a finite cyclic group action on a finite set and a polynomial. Sagan observed the CSP on the set of non-crossing matchings on [2n] := {1, 2, . . . , 2n} using the cyclic group C2n generated by a cyclic shift of order 2n and the q-Catalan polynomial X(q) = 1 [n+1]q [2n n ]q. Bowling-Liang presented a similar result on the set of one-crossing matchings with a completely different proof. We focus on the set Pn of all matchings on [2n] rather than a set of matchings of a particular number of crossings. We find the number of elements in Pn fixed by 2π d rotations, or c 2n/d where c = (1 2 . . . 2n), for d|2n. We find the polynomials Xn(q) such that Pn together with Xn(q) and C2n exhibits the CSP.
منابع مشابه
Cyclic Sieving of Matchings
Let Pn denote the poset of matchings on 2n points on a circle, labeled 1, 2, . . . , 2n in cyclic order, in which we define an element to be less than another element if it can be obtained by an uncrossing. Let Pn,k denote the level of the poset with k crossings. Let Pn,k denote the level of the poset with k crossings. We study the cyclic sieving phenomenon of Pn,k. It is known that the triple ...
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