A007870 -id:A007870 - cp's OEIS frontend

A086644 Permanent of the character table of the symmetric group S_n.

1, 0, 0, 0, 0, 0, 0, -20834715303936, 602706855887546351616

Offset: 1

Yuval Dekel (dekelyuval(AT)hotmail.com), Jul 26 2003

In the Schmidt-Simion reference it is proved that if A000701(n) is odd then a(n) = 0. So a(11)=a(12)=a(13)=0. The computation of a(10) involves a permanent of dimension 42, and it may take a long time for GAP to compute it.

F. W. Schmidt and R. Simion, On a partition identity, J. Combin. Theory, A 36 (1984), 249-252.

Cf. A007870.

GAP
```
Permanent(Irr(SymmetricGroup(n)));
```

a(9) from Ferenc Szollosi, Jul 25 2014

A325536 Sum of sums of omegas of parts over all integer partitions of n.

Original entry on oeis.org

0, 0, 1, 2, 6, 9, 19, 28, 51, 75, 119, 170, 261, 362, 525, 723, 1019, 1373, 1890, 2512, 3386, 4452, 5893, 7658, 10017, 12881, 16627, 21210, 27097, 34266, 43392, 54462, 68399, 85285, 106305, 131712, 163132, 200936, 247332, 303066, 370989, 452296, 550875, 668495

Offset: 0

Gus Wiseman, May 08 2019

nonn

Also omega of the product of products of parts over all integer partitions of n.

The omega of n is A001222(n), the number of prime factors of n counted with multiplicity.

			The integer partitions of 5 are {(5), (4,1), (3,2), (3,1,1), (2,2,1), (2,1,1,1), (1,1,1,1,1)} with products {5,4,6,3,4,2,1} with product 2880 with omega 9, so a(5) = 9.

Cf. A001222, A006128, A006906, A007870, A066186, A066633, A145519, A215366, A325507.

Table[Plus@@PrimeOmega/@Join@@IntegerPartitions[n],{n,0,30}]

a(n) = A001222(A007870(n)).

A346788 Product over all partitions lambda of n of the product of distinct parts in lambda.

Original entry on oeis.org

1, 1, 2, 6, 48, 1440, 414720, 2090188800, 1155790798848000, 226483217146419609600000, 302971317675145105975227187200000000, 37917003542135076706761224377027811868672000000000000, 45800346382799680410294841758069930049013501333211737122406400000000000000000

Offset: 0

Alois P. Heinz, Aug 03 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..20

Cf. A000142, A007870, A162506, A230053, A325504.

a:= n-> mul(i, i=map(x-> {x[]}[], combinat[partition](n))):
seq(a(n), n=0..12);

a[n_] := Times @@ Times @@@ Union /@ IntegerPartitions[n];
a /@ Range[0, 20] (* Jean-François Alcover, Aug 09 2021 *)

a(n) = vecprod(apply(x->vecprod(Set(x)), partitions(n))); \\ Michel Marcus, Aug 04 2021

a(n) = A230053(n) for n <= 6.

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A086644 Permanent of the character table of the symmetric group S_n.

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A325536 Sum of sums of omegas of parts over all integer partitions of n.

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A346788 Product over all partitions lambda of n of the product of distinct parts in lambda.

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