形状为$(n,n,n)$的标准杨表中第一次下降的位置

中文摘要 AI 辅助

设$a(n)$为形状$(n,n,n)$的标准杨表中第2行第1列的条目为奇数的个数；等价地，即第一次下降为偶数的个数。这是整数序列在线百科中的条目A011553，于1996年收录，经过三十年的整理仍无公式。我们给出一个公式，$a(n) = 8\\,\bigl(n!\\,(n+2)!\bigr)^{-1}\sum_{m=1}^{\lfloor n/2\rfloor} m(m+1)(3n-2m-1)!/(n-2m)!$，并用它解决该条目记录的两个猜想。一个创造性的伸缩证书表明$a$满足一个具有多项式系数的二阶线性递推；R. J. Mathar在2023年猜想的三阶递推是它的左倍数，显式余因子为$(4S^{-1}-3)/(7n-9)$。我们还证明$a(n)\sim 3^{3n+7/2}/(64\pi n^{4})$，即V. Kotesovec在2014年猜想的渐近式。第二个证明给出的结果略多于猜想所要求的：第一次下降的位置具有极限分布，$(2,1)$条目等于$r+1$的概率趋于$r(r+2)/3^{r+1}$。对偶数项求和，形状$(n,n,n)$的均匀随机杨表具有奇数$(2,1)$条目的概率趋于$27/64$，而对于形状不受限制的$n$个单元格的均匀随机杨表，相应的极限为$1/e$。

英文摘要

Let $a(n)$ be the number of standard Young tableaux of shape $(n,n,n)$ whose entry in row $2$, column $1$ is odd; equivalently, the number of those whose first descent is even. This is entry A011553 of the On-Line Encyclopedia of Integer Sequences, contributed in 1996, and after thirty years of curation it carries no formula. We supply one, $a(n) = 8\,\bigl(n!\,(n+2)!\bigr)^{-1}\sum_{m=1}^{\lfloor n/2\rfloor} m(m+1)(3n-2m-1)!/(n-2m)!$, and use it to settle both of the conjectures the entry records. A creative telescoping certificate shows that $a$ satisfies a linear recurrence of order two with polynomial coefficients; the order-three recurrence conjectured by R. J. Mathar in 2023 is a left multiple of it, with explicit cofactor $(4S^{-1}-3)/(7n-9)$. We also prove $a(n)\sim 3^{3n+7/2}/(64πn^{4})$, the asymptotic conjectured by V. Kotesovec in 2014. The second proof gives slightly more than the conjecture asks: the position of the first descent has a limiting distribution, the probability that the $(2,1)$ entry equals $r+1$ tending to $r(r+2)/3^{\,r+1}$. Summing the even terms, a uniformly random tableau of shape $(n,n,n)$ has an odd $(2,1)$ entry with probability tending to $27/64$, whereas for a uniformly random tableau of $n$ cells of unrestricted shape the corresponding limit is $1/e$.

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