skip to main content


This content will become publicly available on September 12, 2025

Title: Distributions of Hook lengths in integer partitions

Motivated by the many roles that hook lengths play in mathematics, we study the distribution of the number oftt-hooks in the partitions ofnn. We prove that the limiting distribution is normal with mean\[μ<#comment/>t(n)∼<#comment/>6nπ<#comment/>−<#comment/>t2\mu _t(n)\sim \frac {\sqrt {6n}}{\pi }-\frac {t}{2}\]and variance\[σ<#comment/>t2(n)∼<#comment/>(π<#comment/>2−<#comment/>6)6n2π<#comment/>3.\sigma _t^2(n)\sim \frac {(\pi ^2-6)\sqrt {6n}}{2\pi ^3}.\]Furthermore, we prove that the distribution of the number of hook lengths that are multiples of a fixedt≥<#comment/>4t\geq 4in partitions ofnnconverge to a shifted Gamma distribution with parameterk=(t−<#comment/>1)/2k=(t-1)/2and scaleθ<#comment/>=2/(t−<#comment/>1)\theta =\sqrt {2/(t-1)}.

 
more » « less
PAR ID:
10541502
Author(s) / Creator(s):
; ;
Publisher / Repository:
American Mathematical Society (AMS)
Date Published:
Journal Name:
Proceedings of the American Mathematical Society, Series B
Volume:
11
Issue:
38
ISSN:
2330-1511
Format(s):
Medium: X Size: p. 422-435
Size(s):
p. 422-435
Sponsoring Org:
National Science Foundation
More Like this
  1. Letffbe analytic on[0,1][0,1]with|f(k)(1/2)|⩽<#comment/>Aα<#comment/>kk!|f^{(k)}(1/2)|\leqslant A\alpha ^kk!for some constantsAAandα<#comment/>>2\alpha >2and allk⩾<#comment/>1k\geqslant 1. We show that the median estimate ofμ<#comment/>=∫<#comment/>01f(x)dx\mu =\int _0^1f(x)\,\mathrm {d} xunder random linear scrambling withn=2mn=2^mpoints converges at the rateO(n−<#comment/>clog⁡<#comment/>(n))O(n^{-c\log (n)})for anyc>3log⁡<#comment/>(2)/π<#comment/>2≈<#comment/>0.21c> 3\log (2)/\pi ^2\approx 0.21. We also get a super-polynomial convergence rate for the sample median of2k−<#comment/>12k-1random linearly scrambled estimates, whenk/mk/mis bounded away from zero. Whenffhas app’th derivative that satisfies aλ<#comment/>\lambda-Hölder condition then the median of means has errorO(n−<#comment/>(p+λ<#comment/>)+ϵ<#comment/>)O( n^{-(p+\lambda )+\epsilon })for anyϵ<#comment/>>0\epsilon >0, ifk→<#comment/>∞<#comment/>k\to \inftyasm→<#comment/>∞<#comment/>m\to \infty. The proof techniques use methods from analytic combinatorics that have not previously been applied to quasi-Monte Carlo methods, most notably an asymptotic expression from Hardy and Ramanujan on the number of partitions of a natural number.

     
    more » « less
  2. In this paper we consider which families of finite simple groupsGGhave the property that for eachϵ<#comment/>>0\epsilon > 0there existsN>0N > 0such that, if|G|≥<#comment/>N|G| \ge NandS,TS, Tare normal subsets ofGGwith at leastϵ<#comment/>|G|\epsilon |G|elements each, then every non-trivial element ofGGis the product of an element ofSSand an element ofTT.

    We show that this holds in a strong and effective sense for finite simple groups of Lie type of bounded rank, while it does not hold for alternating groups or groups of the formPSLn(q)\mathrm {PSL}_n(q)whereqqis fixed andn→<#comment/>∞<#comment/>n\to \infty. However, in the caseS=TS=TandGGalternating this holds with an explicit bound onNNin terms ofϵ<#comment/>\epsilon.

    Related problems and applications are also discussed. In particular we show that, ifw1,w2w_1, w_2are non-trivial words,GGis a finite simple group of Lie type of bounded rank, and forg∈<#comment/>Gg \in G,Pw1(G),w2(G)(g)P_{w_1(G),w_2(G)}(g)denotes the probability thatg1g2=gg_1g_2 = gwheregi∈<#comment/>wi(G)g_i \in w_i(G)are chosen uniformly and independently, then, as|G|→<#comment/>∞<#comment/>|G| \to \infty, the distributionPw1(G),w2(G)P_{w_1(G),w_2(G)}tends to the uniform distribution onGGwith respect to theL∞<#comment/>L^{\infty }norm.

     
    more » « less
  3. By discretizing an argument of Kislyakov, Naor and Schechtman proved that the 1-Wasserstein metric over the planar grid{0,1,…<#comment/>,n}2\{0,1,\dots , n\}^2hasL1L_1-distortion bounded below by a constant multiple oflog⁡<#comment/>n\sqrt {\log n}. We provide a new “dimensionality” interpretation of Kislyakov’s argument, showing that if{Gn}n=1∞<#comment/>\{G_n\}_{n=1}^\inftyis a sequence of graphs whose isoperimetric dimension and Lipschitz-spectral dimension equal a common numberδ<#comment/>∈<#comment/>[2,∞<#comment/>)\delta \in [2,\infty ), then the 1-Wasserstein metric overGnG_nhasL1L_1-distortion bounded below by a constant multiple of(log⁡<#comment/>|Gn|)1δ<#comment/>(\log |G_n|)^{\frac {1}{\delta }}. We proceed to compute these dimensions for⊘<#comment/>\oslash-powers of certain graphs. In particular, we get that the sequence of diamond graphs{Dn}n=1∞<#comment/>\{\mathsf {D}_n\}_{n=1}^\inftyhas isoperimetric dimension and Lipschitz-spectral dimension equal to 2, obtaining as a corollary that the 1-Wasserstein metric overDn\mathsf {D}_nhasL1L_1-distortion bounded below by a constant multiple oflog⁡<#comment/>|Dn|\sqrt {\log | \mathsf {D}_n|}. This answers a question of Dilworth, Kutzarova, and Ostrovskii and exhibits only the third sequence ofL1L_1-embeddable graphs whose sequence of 1-Wasserstein metrics is notL1L_1-embeddable.

     
    more » « less
  4. We show that for primesN,p≥<#comment/>5N, p \geq 5withN≡<#comment/>−<#comment/>1modpN \equiv -1 \bmod p, the class number ofQ(N1/p)\mathbb {Q}(N^{1/p})is divisible bypp. Our methods are via congruences between Eisenstein series and cusp forms. In particular, we show that whenN≡<#comment/>−<#comment/>1modpN \equiv -1 \bmod p, there is always a cusp form of weight22and levelΓ<#comment/>0(N2)\Gamma _0(N^2)whoseℓ<#comment/>\ellth Fourier coefficient is congruent toℓ<#comment/>+1\ell + 1modulo a prime abovepp, for all primesℓ<#comment/>\ell. We use the Galois representation of such a cusp form to explicitly construct an unramified degree-ppextension ofQ(N1/p)\mathbb {Q}(N^{1/p}).

     
    more » « less
  5. We develop a higher semiadditive version of Grothendieck-Witt theory. We then apply the theory in the case of a finite field to study the higher semiadditive structure of theK(1)K(1)-local sphereSK(1)\mathbb {S}_{K(1)}at the prime22, in particular realizing the non-22-adic rational element1+ε<#comment/>∈<#comment/>π<#comment/>0SK(1)1+\varepsilon \in \pi _0\mathbb {S}_{K(1)}as a “semiadditive cardinality.” As a further application, we compute and clarify certain power operations inπ<#comment/>0SK(1)\pi _0\mathbb {S}_{K(1)}.

     
    more » « less