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Asymptotic conjecture for parabolic double cosets in symmetric groups

Let SnS_n be the symmetric group, and let pnp_n denote the total number of distinct parabolic double cosets in SnS_n, equivalently

pn=∑w∈Sncw,p_n= \sum_{w\in S_n}c_w,

where cwc_w is the number of parabolic double cosets with minimal element ww. Asymptotic conjecture. There exists a constant KK such that

pnn!∼Klog⁡2n2.\frac{p_n}{n!}\sim\frac{K}{\log^{2n}2}.

This conjecture predicts the asymptotic growth of the total number of parabolic double cosets in the symmetric groups. The preceding results provide an explicit formula for cwc_w and allow computation of the initial values of pnp_n, but no resolution of this asymptotic claim is given.

Equivalent formulations 1

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

  1. Asymptotic conjecture for parabolic double cosets in symmetric groups

    Let pnp_n denote the number of parabolic double cosets in the symmetric group SnS_n. There is a constant KK such that

    pnn!∼K(log⁡2)2n.\frac{p_n}{n!}\sim\frac{K}{(\log 2)^{2n}}.

    Asymptotic conjecture. There exists a constant KK so that

    pnn!∼K(log⁡2)2n.\frac{p_n}{n!}\sim\frac{K}{(\log 2)^{2n}}.

    The paper states that its third main result proves this conjecture, so the claim is solved.

    source: Thomas Browning, “Counting Parabolic Double Cosets in Symmetric Groups”, arXiv:2010.13256 (2021).

Sources & referencesView supporting material

Primary source

Sara C. Billey, Matjaž Konvalinka, T. Kyle Petersen, William Slofstra and Bridget E. Tenner, “Parabolic double cosets in Coxeter groups”, arXiv:1612.00736 (2017).

asymptotic-combinatoricsasymptoticscoxeter-groupsdouble-cosetssymmetric-groups

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