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A318296 Number of conjugacy classes of the Sylow 2-subgroup of the alternating group on n letters. 0
1, 1, 1, 1, 2, 2, 3, 3, 4, 4, 7, 7, 9, 9, 11, 11, 18, 18 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,5

COMMENTS

Also number of partitions of n containing only powers of 2 and having an even number of even elements.

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]

LINKS

Table of n, a(n) for n=1..18.

FORMULA

a(2n) = a(2n+1) for all n. - Charlie Neder, Feb 09 2019

CROSSREFS

Cf. A018819, A000123, A000041.

Sequence in context: A035396 A326077 A005138 * A035679 A326439 A136537

Adjacent sequences:  A318293 A318294 A318295 * A318297 A318298 A318299

KEYWORD

nonn

AUTHOR

Richard Locke Peterson, Aug 23 2018

STATUS

approved

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Last modified August 10 02:28 EDT 2020. Contains 336367 sequences. (Running on oeis4.)