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%I #64 Feb 26 2018 19:23:17
%S 1,1,1,1,1,3,1,1,3,5,1,5,1,7,11,1,1,11,1,13,23,9,1,7,11,11,11,25,1,51,
%T 1,1,39,13,45,23,1,15,59,25,1,135,1,41,73,17,1,9,45,73,83,61,1,45,107,
%U 63,111,19,1,135,1,21,259,1,205,279,1,85,143,349,1
%N Number of normal generalized Young tableaux, of shape the integer partition with Heinz number n, with all rows and columns strictly increasing.
%C A generalized Young tableau of shape y is an array obtained by replacing the dots in the Ferrers diagram of y with positive integers. 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).
%e The a(15) = 11 tableaux:
%e 1 2 3 1 2 4 1 3 4 1 2 5 1 3 5
%e 4 5 3 5 2 5 3 4 2 4
%e .
%e 1 2 3 1 2 3 1 2 4 1 2 4 1 3 4
%e 2 4 3 4 2 3 3 4 2 4
%e .
%e 1 2 3
%e 2 3
%t a[n_]:=If[n===1,1,Sum[a[n/q*Times@@Cases[FactorInteger[q],{p_,k_}:>If[p===2,1,NextPrime[p,-1]^k]]],{q,Select[Rest[Divisors[n]],SquareFreeQ]}]];
%t Array[a,100]
%Y Cf. A000085, A001221, A005117, A006958, A015128, A056239, A138178, A153452, A238690, A296150, A296188, A297388, A299925, A299926, A299968, A300118, A300120, A300122.
%K nonn
%O 1,6
%A _Gus Wiseman_, Feb 26 2018
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