Number of permutations with same peak set for signed permutations
Number of permutations with same peak set for signed permutations
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有符号排列具有相同峰值集的排列数
DOI：
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发表时间：
2013
期刊：
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通讯作者：
Rita Zevallos
中科院分区：
文献类型：
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作者：
F. Castro;Alexander Diaz;R. Orellana;Jose Pastrana;Rita Zevallos
高八面体群B_n中的有符号排列pi = pi_1pi_2 ldots pi_n是这样一个词：{-n， ldots, - 1,1,ldots，n}和{|pi_1|， |pi_2|, ldots， |pi_n|}中的每个pi_i = {1,2,ldots,n}。如果pi_{i-1} pi_{i+1}，且P_B（pi）表示pi的所有峰的集合，则索引i是pi的一个峰。给定任意集合S，我们定义P_B（S,n）为B_n中有符号排列pi的集合，且P_B(pi) = S。本文研究了集合P_B（S,n）的基数性。2012年，Billey， burzy和Sagan研究了对称群S_n中置换的类似问题。本文将其结果推广到高八面体群；特别地，我们证明了#P_B(S,n) = p(n)2^{2n-|S|-1}，其中p(n)是由Billey， burzy和Sagan发现的相同的多项式，这导致了多项式p(n)有趣的特殊情况的显式计算。此外，我们还将这些结果扩展到我们在排列开始时添加pi_0=0的情况，这使得对称基团和高八面体基团在位置1处出现峰值的可能性。
A signed permutation pi = pi_1pi_2 ldots pi_n in the hyperoctahedral group B_n is a word such that each pi_i in {-n, ldots, -1, 1, ldots, n} and {|pi_1|, |pi_2|, ldots, |pi_n|} = {1,2,ldots,n}. An index i is a peak of pi if pi_{i-1} pi_{i+1} and P_B(pi) denotes the set of all peaks of pi. Given any set S, we define P_B(S,n) to be the set of signed permutations pi in B_n with P_B(pi) = S. In this paper we are interested in the cardinality of the set P_B(S,n). In 2012, Billey, Burdzy and Sagan investigated the analogous problem for permutations in the symmetric group, S_n. In this paper we extend their results to the hyperoctahedral group; in particular we show that #P_B(S,n) = p(n)2^{2n-|S|-1} where p(n) is the same polynomial found in by Billey, Burdzy and Sagan which leads to the explicit computation of interesting special cases of the polynomial p(n). In addition we have extended these results to the case where we add pi_0=0 at the beginning of the permutations, which gives rise to the possibility of a peak at position 1, for both the symmetric and the hyperoctahedral groups.