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A007716
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Number of polynomial symmetric functions of matrix of order n under separate row and column permutations.
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419
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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
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OFFSET
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0,3
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COMMENTS
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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
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LINKS
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Andrew Howroyd, Table of n, a(n) for n = 0..50 (terms 0..30 from Seiichi Manyama)
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FORMULA
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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
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EXAMPLE
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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}}.
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MATHEMATICA
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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-François Alcover, May 03 2019, after Andrew Howroyd *)
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PROG
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(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
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CROSSREFS
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Main diagonal of A318795.
Cf. A053307, A052365, A052366, A052367, A052372, A052373, A049311, A054688, A000041.
Sequence in context: A052367 A052372 A052373 * A122948 A317800 A357799
Adjacent sequences: A007713 A007714 A007715 * A007717 A007718 A007719
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KEYWORD
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nice,nonn
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AUTHOR
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Colin Mallows
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EXTENSIONS
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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
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STATUS
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approved
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