A296188 Number of normal semistandard Young tableaux whose shape is the integer partition with Heinz number n.

1, 1, 2, 1, 4, 4, 8, 1, 6, 12, 16, 6, 32, 32, 28, 1, 64, 16, 128, 24, 96, 80, 256, 8, 44, 192, 22, 80, 512, 96, 1024, 1, 288, 448, 224, 30, 2048, 1024, 800, 40, 4096, 400, 8192, 240, 168, 2304, 16384, 10, 360, 204, 2112, 672, 32768, 68, 832, 160, 5376, 5120

Offset

1,3

Comments

A tableau is normal if its entries span an initial interval of positive integers. The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

References

Richard P. Stanley, Enumerative Combinatorics Volume 2, Cambridge University Press, 1999, Chapter 7.10.

Links

Table of n, a(n) for n=1..58.

FindStat - Combinatorial Statistic Finder, Semistandard Young tableaux

Formula

Let b(n) = Sum_{d|n, d>1} b(n * d' / d) where if d = Product_i prime(s_i)^m(i) then d' = Product_i prime(s_i - 1)^m(i) and prime(0) = 1. Then a(n) = b(conj(n)) where conj = A122111.

Example

The a(9) = 6 tableaux:
1 3   1 2   1 2   1 2   1 1   1 1
2 4   3 4   3 3   2 3   2 3   2 2

Mathematica

conj[y_List]:=If[Length[y]===0, y, Table[Length[Select[y, #>=k&]], {k, 1, Max[y]}]];
conj[n_Integer]:=Times@@Prime/@conj[If[n===1, {}, Join@@Cases[FactorInteger[n]//Reverse, {p_, k_}:>Table[PrimePi[p], {k}]]]];
ssyt[n_]:=If[n===1, 1, Sum[ssyt[n/q*Times@@Cases[FactorInteger[q], {p_, k_}:>If[p===2, 1, NextPrime[p, -1]^k]]], {q, Rest[Divisors[n]]}]];
Table[ssyt[conj[n]], {n, 50}]

Crossrefs

Cf. A000085, A001222, A056239, A063834, A112798, A122111, A138178, A153452, A191714, A210391, A228125, A296150, A296560, A296561, A299202, A299966, A300056, A300121.

Sequence in context: A339407 A077965 A077967 * A008312 A345442 A060723

Adjacent sequences: A296185 A296186 A296187 * A296189 A296190 A296191

Keyword

nonn

Author

Gus Wiseman, Feb 14 2018

Status

approved