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A007716 Number of polynomial symmetric functions of matrix of order n under separate row and column permutations. 407
1, 1, 4, 10, 33, 91, 298, 910, 3017, 9945, 34207, 119369, 429250, 1574224, 5916148, 22699830, 89003059, 356058540, 1453080087, 6044132794, 25612598436, 110503627621, 485161348047, 2166488899642, 9835209912767, 45370059225318, 212582817739535, 1011306624512711 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Also, the number of nonnegative integer n X n matrices with sum of elements equal to n, under row and column permutations (cf. A120733).

This is a two-dimensional generalization of the partition function (A000041), which equals the number of length n vectors of nonnegative integers with sum n, equivalent under permutations. - Franklin T. Adams-Watters, Sep 19 2011

Also number of non-isomorphic multiset partitions of weight n. - Gus Wiseman, Sep 19 2011

LINKS

Andrew Howroyd, Table of n, a(n) for n = 0..50 (terms 0..30 from Seiichi Manyama)

FORMULA

a(n) is the coefficient of x^n in the cycle index Z(S_n X S_n; x_1, x_2, ...) if we replace x_i with 1+x^i+x^(2*i)+x^(3*i)+x^(4*i)+..., where S_n X S_n is the Cartesian product of symmetric groups S_n of degree n. - Vladeta Jovovic, Mar 09 2000

EXAMPLE

The 10 non-isomorphic multiset partitions of weight 3 are {{1, 1, 1}}, {{1, 1, 2}}, {{1, 2, 3}}, {{1}, {1, 1}}, {{1}, {1, 2}}, {{1}, {2, 2}}, {{1}, {2, 3}}, {{1}, {1}, {1}}, {{1}, {1}, {2}}, {{1}, {2}, {3}}.

MATHEMATICA

permcount[v_] := Module[{m = 1, s = 0, k = 0, t}, For[i = 1, i <= Length[v], i++, t = v[[i]]; k = If[i>1 && t == v[[i-1]], k+1, 1]; m *= t*k; s += t]; s!/m];

c[p_, q_, k_] := SeriesCoefficient[1/Product[(1-x^LCM[p[[i]], q[[j]]])^GCD[ p[[i]], q[[j]]], {j, 1, Length[q]}, {i, 1, Length[p]}], {x, 0, k}];

M[m_, n_, k_] := Module[{s=0}, Do[Do[s += permcount[p]*permcount[q]*c[p, q, k], {q, IntegerPartitions[n]}], {p, IntegerPartitions[m]}]; s/(m!*n!)];

a[n_] := a[n] = M[n, n, n];

Table[Print[n, " ", a[n]]; a[n], {n, 0, 18}] (* Jean-Fran├žois Alcover, May 03 2019, after Andrew Howroyd *)

PROG

(PARI) \\ See A318795

a(n) = M(n, n, n); \\ Andrew Howroyd, Sep 03 2018

(PARI)

EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}

permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}

K(q, t, k)={EulerT(Vec(sum(j=1, #q, gcd(t, q[j])*x^lcm(t, q[j])) + O(x*x^k), -k))}

a(n)={my(s=0); forpart(q=n, s+=permcount(q)*polcoef(exp(x*Ser(sum(t=1, n, K(q, t, n)/t))), n)); s/n!} \\ Andrew Howroyd, Mar 29 2020

CROSSREFS

Main diagonal of A318795.

Cf. A053307, A052365, A052366, A052367, A052372, A052373, A049311, A054688, A000041.

Sequence in context: A052367 A052372 A052373 * A122948 A317800 A149171

Adjacent sequences:  A007713 A007714 A007715 * A007717 A007718 A007719

KEYWORD

nice,nonn

AUTHOR

Colin Mallows

EXTENSIONS

More terms from Vladeta Jovovic, Jun 28 2000

a(19)-a(25) from Max Alekseyev, Jan 22 2010

a(0)=1 prepended by Alois P. Heinz, Feb 03 2019

a(26)-a(27) from Seiichi Manyama, Nov 23 2019

STATUS

approved

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Last modified October 25 15:37 EDT 2020. Contains 338012 sequences. (Running on oeis4.)