Counting Permutations in S_2n and S_2n+1

2024/07/10 by Yuewen Luo, Luo, Yuewen
Computer Science · Engineering · Mathematics · #Advanced Combinatorial Mathematics #Algorithms and Data Compression #Combinatorics (math.CO) #FOS: Mathematics #graph theory and CDMA systems

paper · pdf · doi:10.48550/arxiv.2407.07366

openalex publication_date 2024/07/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let α(n) denote the number of perfect square permutations in the symmetric group S_n. The conjecture α(2n+1) = (2n+1) α(2n), provided by Stanley[4], was proved by Blum[1] using a generating function. This paper presents a combinatorial proof for this conjecture. At the same time, we demonstrate that all permutations with an even number of even cycles in both S_2n and S_2n+1 can be categorized into three distinct types that correspond to each other.

Related

• A note on a question due to A. Garsia
• Descent set distribution for permutations with cycles of only odd or only even lengths
• Number of permutations with same peak set for signed permutations
• Counting occurences of 132 in a permutation
• The Probability of Generating the Symmetric Group