|
%I #12 Feb 09 2019 23:29:48
%S 1,1,1,1,2,2,3,3,4,4,7,7,9,9,11,11,18,18
%N Number of conjugacy classes of the Sylow 2-subgroup of the alternating group on n letters.
%C Also number of partitions of n containing only powers of 2 and having an even number of even elements.
%C These partitions form a semiring. The semiring uses the following binary operations *,+: Let A=(a1,a2,..,aj) be a partition of k that has j parts with i of those j being powers of 2 greater than 1, written in nonincreasing order. Let B be a partition of y that has x parts, with w of the x being powers of 2 greater than 1, arranged in descending order. Then A+B = (a1,a2,...,aj,b1,b2,...,bx), and A*B=AB=(a1,a2,...,aj)*(b1,b2,...,bx) is defined to be the partition (a1b1,a2b1,...,a1bx,a2b1,...,a2bx,...,ajbx) of ky. Since i and w are even by assumption, the numbers of powers of two in A+B (= i + w) and AB (= ix + jw - iw) must also be even, and both are members of the semiring. In addition, if C = (c1,...cm) is a partition of k into m parts, n of which are powers of two, (AB)C = A(BC) = (a1b1c1,a2b1c1,...,ajbxcm), and (A+B)C = (a1,a2,...,aj,b1,b2,...,bx)(c1,...cm) = (a1c1,a2c1,...,ajcm,b1c1,...) = (a1c1,a2c1,...,ajcm) + (b1c1,b2c1,...,bxcm) = AC + BC, so the necessary criteria for a semiring hold. [Missing parts added by _Charlie Neder_, Feb 09 2019]
%F a(2n) = a(2n+1) for all n. - _Charlie Neder_, Feb 09 2019
%Y Cf. A018819, A000123, A000041.
%K nonn
%O 1,5
%A _Richard Locke Peterson_, Aug 23 2018
|
