The first descent in a standard Young tableau of shape (n,n,n)
Abstract
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 (n! (n+2)!)−1∑m=1⌊ n/2⌋ 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)∼ 33n+7/2/(64πn4), 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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BibTeX
@misc{niu2026first,
title = {{The first descent in a standard Young tableau of shape $(n,n,n)$}},
author = {Tong Niu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.03038},
eprint = {2610.03038},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.03038},
}
RIS
TY - UNPB
TI - The first descent in a standard Young tableau of shape (n,n,n)
AU - Tong Niu
PY - 2026
DA - 2026/10/02
PB - arXiv
DO - 10.48550/arxiv.2610.03038
UR - https://arxiv.org/abs/2610.03038
AB - 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 (n! (n+2)!)^−1∑_m=1^⌊ n/2⌋ 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)∼ 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.
ER -