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.
Original entry on oeis.org
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
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
Cf.
A000085,
A000219,
A003293,
A114736,
A138178,
A299968,
A323436,
A323437,
A323438,
A323439,
A323581.
-
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}]
A323581
Number of ways to fill a Young diagram with positive integers summing to n such that the rows are strictly increasing and the columns are strictly decreasing.
Original entry on oeis.org
1, 1, 1, 3, 3, 5, 8, 10, 14, 19, 28, 34, 48, 60, 80, 106, 134, 171, 222, 279, 354, 452, 562, 706, 884, 1100
Offset: 0
The a(8) = 14 tableaux:
8 1 7 2 6 3 5 1 2 5 1 3 4
.
7 6 5 2 5 3 4 2 3
1 2 3 1 1 1 2
.
5 4
2 3
1 1
Cf.
A000085,
A000219,
A003293,
A114736,
A138178,
A299968,
A323436,
A323437,
A323438,
A323439,
A323580.
-
primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
sqfacs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[sqfacs[n/d],Min@@#>=d&]],{d,Select[Rest[Divisors[n]],SquareFreeQ]}]];
Table[Sum[Length[Select[Reverse/@Sort/@Map[primeMS,sqfacs[y],{2}],And@@Greater@@@DeleteCases[Transpose[PadRight[#]],0,{2}]&]],{y,Times@@Prime/@#&/@IntegerPartitions[n]}],{n,10}]