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

%I A296188 #20 Feb 27 2018 20:01:40
%S A296188 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,
%T A296188 22,80,512,96,1024,1,288,448,224,30,2048,1024,800,40,4096,400,8192,
%U A296188 240,168,2304,16384,10,360,204,2112,672,32768,68,832,160,5376,5120
%N A296188 Number of normal semistandard Young tableaux whose shape is the integer partition with Heinz number n.
%C A296188 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).
%D A296188 Richard P. Stanley, Enumerative Combinatorics Volume 2, Cambridge University Press, 1999, Chapter 7.10.
%H A296188 FindStat - Combinatorial Statistic Finder, <a href="http://www.findstat.org/SemistandardTableaux">Semistandard Young tableaux</a>
%F A296188 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.
%e A296188 The a(9) = 6 tableaux:
%e A296188 1 3   1 2   1 2   1 2   1 1   1 1
%e A296188 2 4   3 4   3 3   2 3   2 3   2 2
%t A296188 conj[y_List]:=If[Length[y]===0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
%t A296188 conj[n_Integer]:=Times@@Prime/@conj[If[n===1,{},Join@@Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A296188 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]]}]];
%t A296188 Table[ssyt[conj[n]],{n,50}]
%Y A296188 Cf. A000085, A001222, A056239, A063834, A112798, A122111, A138178, A153452, A191714, A210391, A228125, A296150, A296560, A296561, A299202, A299966, A300056, A300121.
%K A296188 nonn
%O A296188 1,3
%A A296188 _Gus Wiseman_, Feb 14 2018