Year: 2024
Type: article
Abstract: Let $$S_n$$ S n denote the symmetric group of permutations acting on n elements. We investigate the double sequence $$\{N_{\ell }(n)\}$$ { N ℓ ( n ) } counting the number of $$\ell $$ ℓ tuples of elements of the symmetric group $$S_n$$ S n , where the components commute, normalized by the order of $$S_n$$ S n . Our focus lies on exploring log-concavity with respect to n: $$\begin{aligned} N_{\ell }(n)^2 - N_{\ell }(n-1) \,\, N_{\ell }(n+1) \ge 0. \end{aligned}$$ N ℓ ( n ) 2 - N ℓ ( n - 1 ) N ℓ ( n + 1 ) ≥ 0 . We establish that this depends on $$n \pmod {3}$$ n ( mod 3 ) for sufficiently large $$\ell $$ ℓ . These numbers are studied by Bryan and Fulman as the nth orbifold characteristics, generalizing work by Macdonald and Hirzebruch–Hofer concerning the ordinary and string-theoretic Euler characteristics of symmetric products. Notably, $$N_2(n)$$ N 2 ( n ) represents the partition numbers p(n), while $$N_{3}(n)$$ N 3 ( n ) represents the number of non-equivalent n-sheeted coverings of a torus studied by Liskovets and Medynkh. The numbers also appear in algebra since $$ \vert S_n \vert \,\, N_{\ell }(n) = \left| \textrm{Hom}\left( \mathbb {Z}^{\ell },S_n\right) \right| $$ | S n | N ℓ ( n ) = Hom Z ℓ , S n
Source: The Ramanujan Journal
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $2,890
Funder: Universität zu Köln
OpenAlex ID: W4405740611