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.
1, 1, 3, 6, 13, 23, 45, 76, 136, 225, 381, 611, 1001, 1570, 2489, 3842, 5948, 9022, 13714, 20501, 30649, 45262, 66721, 97393, 141888, 204993
Offset: 0
Examples
The a(5) = 23 tableaux: 5 41 32 311 221 2111 11111 . 1 2 11 21 11 111 111 1111 4 3 3 2 21 2 11 1 . 1 1 11 11 111 1 2 1 11 1 3 2 2 1 1 . 1 11 1 1 1 1 2 1 . 1 1 1 1 1
Links
- nLab, Young Diagram.
Crossrefs
Programs
-
Mathematica
primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]; facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]]; Table[Sum[Length[Select[Reverse/@Sort/@Map[primeMS,facs[y],{2}],And@@(GreaterEqual@@@Transpose[PadRight[#]])&]],{y,Times@@Prime/@#&/@IntegerPartitions[n]}],{n,10}]