cp's OEIS Frontend

A299966 Number of normal generalized Young tableaux, of shape the integer partition with Heinz number n, with all rows and columns weakly increasing and all regions non-singleton skew-partitions.

%I A299966 #9 Feb 23 2018 11:12:00
%S A299966 1,0,1,1,1,1,2,1,3,3,3,3,5,5,5,2,8,5,13,6,13,10,21,5,11,18,11,14,34,
%T A299966 15,55,3,26,33,23,13,89,59,54,14,144,38,233,28,31,105,377,10,47,31,
%U A299966 106,57,610,23,60,32,206,185,987,38,1597,324,91,5,132,93,2584,111
%N A299966 Number of normal generalized Young tableaux, of shape the integer partition with Heinz number n, with all rows and columns weakly increasing and all regions non-singleton skew-partitions.
%C A299966 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).
%D A299966 Bruce E. Sagan, The Symmetric Group, Springer-Verlag New York, 2001.
%H A299966 Gus Wiseman, <a href="/A299966/a299966.png">The a(30) = 15 non-singleton tableaux of shape (321).</a>
%e A299966 The a(25) = 11 tableaux:
%e A299966 1 2 3   1 2 2   1 1 3   1 1 2
%e A299966 1 2 3   1 3 3   2 2 3   2 3 3
%e A299966 .
%e A299966 1 2 2   1 1 2   1 1 2   1 1 2   1 1 1   1 1 1
%e A299966 1 2 2   2 2 2   1 2 2   1 1 2   2 2 2   1 2 2
%e A299966 .
%e A299966 1 1 1
%e A299966 1 1 1
%t A299966 primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A299966 undptns[y_]:=DeleteCases[Select[Tuples[Range[0,#]&/@y],OrderedQ[#,GreaterEqual]&],0,{2}];
%t A299966 eh[y_]:=If[Total[y]=!=1,1,0]+Sum[eh[c],{c,Select[undptns[y],Total[#]>1&&Total[y]-Total[#]>1&]}];
%t A299966 Table[eh[Reverse[primeMS[n]]],{n,60}]
%Y A299966 Cf. A000085, A056239, A063834, A112798, A122111, A138178, A153452, A238690, A296150, A296188, A297388, A299925, A299926, A299967.
%K A299966 nonn
%O A299966 1,7
%A A299966 _Gus Wiseman_, Feb 22 2018