A321413
Number of non-isomorphic self-dual multiset partitions of weight n with no singletons and relatively prime part sizes.
Original entry on oeis.org
1, 0, 0, 0, 0, 3, 0, 14, 13, 50, 65
Offset: 0
Non-isomorphic representatives of the a(5) = 3, a(7) = 14, and a(8) = 13 multiset partitions:
{{11}{122}} {{111}{1222}} {{111}{11222}}
{{11}{222}} {{111}{2222}} {{111}{22222}}
{{12}{122}} {{112}{1222}} {{112}{12222}}
{{11}{22222}} {{122}{11222}}
{{12}{12222}} {{11}{122}{233}}
{{122}{1122}} {{11}{122}{333}}
{{22}{11222}} {{11}{222}{333}}
{{11}{12}{233}} {{11}{223}{233}}
{{11}{22}{233}} {{12}{122}{333}}
{{11}{22}{333}} {{12}{123}{233}}
{{11}{23}{233}} {{13}{112}{233}}
{{12}{12}{333}} {{13}{122}{233}}
{{12}{13}{233}} {{23}{123}{123}}
{{13}{23}{123}}
A321412
Number of non-isomorphic self-dual multiset partitions of weight n with no singletons and with aperiodic parts.
Original entry on oeis.org
1, 0, 0, 0, 1, 1, 3, 4, 12, 20, 42
Offset: 0
Non-isomorphic representatives of the a(5) = 1 through a(8) = 12 multiset partitions:
{{12}{12}} {{12}{122}} {{112}{122}} {{112}{1222}} {{1112}{1222}}
{{12}{1222}} {{12}{12222}} {{112}{12222}}
{{12}{13}{23}} {{12}{13}{233}} {{12}{122222}}
{{13}{23}{123}} {{122}{11222}}
{{12}{123}{233}}
{{12}{13}{2333}}
{{13}{112}{233}}
{{13}{122}{233}}
{{13}{23}{1233}}
{{23}{123}{123}}
{{12}{12}{34}{34}}
{{12}{13}{24}{34}}
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}]
A321484
Number of non-isomorphic self-dual connected multiset partitions of weight n.
Original entry on oeis.org
1, 1, 1, 2, 3, 6, 9, 20, 35, 78, 141
Offset: 0
Non-isomorphic representatives of the a(1) = 1 through a(6) = 9 multiset partitions:
{{1}} {{11}} {{111}} {{1111}} {{11111}} {{111111}}
{{2}{12}} {{12}{12}} {{11}{122}} {{112}{122}}
{{2}{122}} {{12}{122}} {{12}{1222}}
{{2}{1222}} {{2}{12222}}
{{2}{13}{23}} {{22}{1122}}
{{3}{3}{123}} {{12}{13}{23}}
{{2}{13}{233}}
{{3}{23}{123}}
{{3}{3}{1233}}
Cf.
A007718,
A056156,
A138178,
A316983,
A319565,
A319616,
A319647,
A319719,
A321194,
A321585,
A321680,
A321681.
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