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A323580 Number of ways to fill a Young diagram with positive integers summing to n such that the rows are weakly decreasing and the columns are weakly increasing.

%I A323580 #6 Jan 20 2019 08:55:55
%S A323580 1,1,3,6,13,23,45,76,136,225,381,611,1001,1570,2489,3842,5948,9022,
%T A323580 13714,20501,30649,45262,66721,97393,141888,204993
%N A323580 Number of ways to fill a Young diagram with positive integers summing to n such that the rows are weakly decreasing and the columns are weakly increasing.
%H A323580 nLab, <a href="https://ncatlab.org/nlab/show/Young+diagram">Young Diagram</a>.
%e A323580 The a(5) = 23 tableaux:
%e A323580   5   41   32   311   221   2111   11111
%e A323580 .
%e A323580   1   2   11   21   11   111   111   1111
%e A323580   4   3   3    2    21   2     11    1
%e A323580 .
%e A323580   1   1   11   11   111
%e A323580   1   2   1    11   1
%e A323580   3   2   2    1    1
%e A323580 .
%e A323580   1   11
%e A323580   1   1
%e A323580   1   1
%e A323580   2   1
%e A323580 .
%e A323580   1
%e A323580   1
%e A323580   1
%e A323580   1
%e A323580   1
%t A323580 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A323580 facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t A323580 Table[Sum[Length[Select[Reverse/@Sort/@Map[primeMS,facs[y],{2}],And@@(GreaterEqual@@@Transpose[PadRight[#]])&]],{y,Times@@Prime/@#&/@IntegerPartitions[n]}],{n,10}]
%Y A323580 Cf. A000085, A000219, A003293, A114736, A138178, A299968, A323436, A323437, A323438, A323439, A323581.
%K A323580 nonn,more
%O A323580 0,3
%A A323580 _Gus Wiseman_, Jan 18 2019