cp's OEIS Frontend

A207542 Number of solid standard Young tableaux with n cells.

%I A207542 #74 Sep 04 2025 00:51:27
%S A207542 1,1,3,9,33,135,633,3207,17589,102627,636033,4161141,28680717,
%T A207542 207318273,1567344549,12345147705,101013795753,856212871761,
%U A207542 7501911705747,67815650852235,631574151445665,6051983918989833,59605200185016639,602764245172225251,6252962956009863363
%N A207542 Number of solid standard Young tableaux with n cells.
%C A207542 A solid standard Young tableaux (SSYT) with n cells is a way of placing the integers from 1 to n in a 3D Young diagram of a plane partition with the property that the entries increase from left to right, back to front, and bottom to top.
%C A207542 It is also the number of almost topological sequences (ATS) for the set N^3 at depth n with (N=set of nonnegative integers). See Balakrishnan et al. for definition and a proof of the bijection between SSYT and ATS. - _Suresh Govindarajan_, Mar 02 2012
%C A207542 Also the number of paths from a plane partition to the empty partition by repeated trimming. - _Wouter Meeussen_, Sep 03 2025
%H A207542 Shalosh B. Ekhad, Doron Zeilberger, and Vaclav Kotesovec, <a href="/A207542/b207542.txt">Table of n, a(n) for n = 0..37</a> (terms 0..30 from Shalosh B. Ekhad and Doron Zeilberger)
%H A207542 S. Balakrishnan, S. Govindarajan, and N. S. Prabhakar, <a href="https://arxiv.org/abs/1105.6231">On the asymptotics of higher-dimensional partitions</a>, J. Phys. A45 (2012) 055001; arXiv:1105.6231 [cond-mat.stat-mech], 2011.
%H A207542 Shalosh B. Ekhad and Doron Zeilberger, <a href="http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/ssyt.html">Computational and Theoretical Challenges on Counting Solid Standard Young Tableaux</a>.
%H A207542 Suresh Govindarajan, <a href="http://boltzmann.wikidot.com/topological-sequences">Almost Topological Sequences</a>
%t A207542 b[n_, k_, L_] := b[n, k, L] = If[n == 0, 1, b[n - 1, k, Append[L, {1}]] + Sum[If[i == 1 || Length[L[[i]]] < Length[L[[i - 1]]], b[n - 1, k, ReplacePart[L, i -> Append[L[[i]], 1]]], 0] + Sum[If[L[[i, j]] < k && (i == 1 || L[[i, j]] < L[[i - 1, j]]) && (j == 1 || L[[i, j]] < L[[i, j - 1]]), b[n - 1, k, ReplacePart[L, i -> ReplacePart[L[[i]], j -> L[[i, j]] + 1]]], 0], {j, 1, Length[L[[i]]]}], {i, 1, Length[L]}]];
%t A207542 A[n_, k_] := If[k == 0, If[n == 0, 1, 0], b[n, Min[n, k], {}]];
%t A207542 T[n_, k_] := A[n, k] - If[k == 0, 0, A[n, k - 1]];
%t A207542 a[n_] := a[n] = Sum[T[n, k], {k, 0, n}];
%t A207542 Table[Print[n, " ", a[n]]; a[n], {n, 0, 20}] (* _Jean-François Alcover_, Apr 28 2022, after _Alois P. Heinz_ in A214753 *)
%Y A207542 Rows sums of A214753.
%Y A207542 Main diagonal of A215086.
%Y A207542 Column k=0 of A215120. - _Alois P. Heinz_, May 12 2014
%K A207542 nonn,hard,changed
%O A207542 0,3
%A A207542 _Matthew C. Russell_, Feb 24 2012