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The notion of rigid commutators is introduced to determine the sequence of the logarithms of the indices of a certain normalizer chain in the Sylow 2-subgroup of the symmetric group on $$2^n$$ 2 n letters. The terms of this sequence are proved to be those of the partial sums of the partitions of an integer into at least two distinct parts, that relates to a famous Euler’s partition theorem.

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“…where the commutators of Eqs. (14) and (15) are respectively obtained by lifting those of Eqs. (12) and (13), and the commutators of Eqs.…”

Section: A Computational Supplementmentioning

confidence: 99%

“…The sequence b j + 1 appears in several others areas of mathematics, from number theory to commutative algebra[14]. In particular, it was already known to Euler that b j + 1 corresponds to the number of partitions of j into odd parts (see[15, Chapter 16] and[3, §3]).…”

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confidence: 99%

Rigid commutators and a normalizer chain

et al. 2021

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“…It is well known [2] that p(n) is asymptotic to 1 4n √ 3 e π √ 2n/3 for n large, whence our bound easily follows. (Up to polynomial factors, this asymptotically improves by an exponent of √ 2 the bound L(n) ≥ q(n) shown in [1], where q(n) denotes the number of distinct-part partitions of n, which required a substantial amount of work. Our bound is also sharper for each n, since p(n − 1) ≥ q(n), with strict inequality for n ≥ 4.…”

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confidence: 93%

A note on the asymptotics of the number of O-sequences of given length

Stanley

Zanello

2019

Discrete Mathematics

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We look at the number L(n) of O-sequences of length n. Recall that an Osequence can be defined algebraically as the Hilbert function of a standard graded k-algebra, or combinatorially as the f -vector of a multicomplex. The sequence L(n) was first investigated in a recent paper by commutative algebraists Enkosky and Stone, inspired by Huneke. In this note, we significantly improve both of their upper and lower bounds, by means of a very short partition-theoretic argument. In particular, it turns out that, for suitable positive constants c 1 and c 2 and all n > 2, e c1 √ n ≤ L(n) ≤ e c2 √ n log n .It remains an open problem to determine an exact asymptotic estimate for L(n).

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“…[2] For all ≥ 1, ( ) ⊆ ( ). In particular the sequence of the cardinality of ( ) is bounded above by the Fibonacci sequence.…”

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confidence: 99%

“…Hilbert functions of graded rings are more convenient for many applications and are known to relate to many different subjects such as dimensions, multiplicity and Betti numbers (see: Bruns and Herzog, [1]). In [2], Enkoskoy and Stone introduced recursion formulas related to Hilbert functions. They showed the term of sequence, whose term is the number of ℎvectors of length , is bounded above by the Fibonacci number.…”

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confidence: 99%