On the polynomiality and asymptotics of moments of sizes for random (n, dn ± 1)-core partitions with distinct parts

NEW

Problem: For fixed $d\in \mathbb Z_{>0}$, let $\mathcal C_{n,d}$ be the set of partitions into distinct parts that are simultaneously $n$-core and $(dn-1)$-core. Choose $\lambda\in\mathcal C_{n,d}$ uniformly and let

The reconstructed conjecture is:

for every fixed $d\ge1$. The word “distribution” is ambiguous literally, but the surrounding paper title and moment context indicate the distribution of the size statistic.

Result: The conjecture is true.

Use the beta-set $\beta(\lambda)=\{\lambda_i+\ell-i:1\le i\le \ell\}$. A partition is strict iff $\beta(\lambda)$ has no two consecutive integers, and an $(s,t)$-core corresponds to an order ideal in Anderson’s poset $P_{s,t}$ of positive integers not representable as $as+bt$.

For $s=n,t=dn-1$,

Writing an order ideal by column heights $h_j$, strictness forces no adjacent positive heights. The core condition then gives exactly

Thus $\mathcal C_{n,d}$ is in bijection with such height sequences.

For $L=\sum h_j$,

Now compare the uniform law on these height sequences with the stationary finite-state Markov chain on $\{0,1,\dots,d\}$ with transition matrix

where $\rho=(1+\sqrt{1+4d})/2$. The stationary law is

The actual finite uniform model differs only by a bounded endpoint tilt and by the boundary restriction $h_1\ne d$, so standard finite-state Markov-chain CLTs apply unchanged.

Let

and define

Then

The weighted finite-state Markov CLT gives convergence to a centered normal law with variance

where $\Gamma(g)$ is the usual long-run variance of $g(H_j)$. This variance is positive: the chain regenerates at state $0$; if $\Gamma(g)=0$, then cycle sums force $g(0)=g(1)=\cdots=g(d)=0$. But $f_t$ is not identically zero for a.e. $t\in[0,1]$. Hence

and therefore

Verification audit: the proof treats all fixed $d\ge1$, including $d=1$; small-$n$ degeneracies do not matter because $\operatorname{Var}(X_{n,d})\sim\sigma_d^2n^3>0$. The boundary condition $h_1\le d-1$ is included, and no extra hypotheses are introduced.

Citation: The exact conjecture is not known to me as previously proved. Standard ingredients used: Anderson, “Partitions which are simultaneously $t_1$- and $t_2$-core,” Discrete Math. 248 (2002), 237–243; and the classical finite-state Markov-chain CLT/Perron–Frobenius perturbation method.