A052365 -id:A052365 - cp's OEIS frontend

A007716 Number of polynomial symmetric functions of matrix of order n under separate row and column permutations.

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

Offset: 0

Colin Mallows

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

			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}}.

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

Main diagonal of A318795.

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

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 *)

\\ See A318795
a(n) = M(n,n,n); \\ Andrew Howroyd, Sep 03 2018

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

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

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

A318795 Array read by antidiagonals: T(n,k) is the number of inequivalent nonnegative integer n X n matrices with sum of elements equal to k, under row and column permutations.

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 5, 4, 1, 1, 11, 10, 4, 1, 1, 14, 24, 10, 4, 1, 1, 24, 51, 33, 10, 4, 1, 1, 30, 114, 78, 33, 10, 4, 1, 1, 45, 219, 224, 91, 33, 10, 4, 1, 1, 55, 424, 549, 277, 91, 33, 10, 4, 1, 1, 76, 768, 1403, 792, 298, 91, 33, 10, 4, 1, 1, 91, 1352, 3292, 2341, 881, 298, 91, 33, 10, 4, 1

Offset: 1

Andrew Howroyd, Sep 03 2018

			Array begins:
===========================================================
n\k| 1 2  3  4  5   6   7    8    9    10     11     12
---+-------------------------------------------------------
1  | 1 1  1  1  1   1   1    1    1     1      1      1 ...
2  | 1 4  5 11 14  24  30   45   55    76     91    119 ...
3  | 1 4 10 24 51 114 219  424  768  1352   2278   3759 ...
4  | 1 4 10 33 78 224 549 1403 3292  7677  16934  36581 ...
5  | 1 4 10 33 91 277 792 2341 6654 18802  51508 138147 ...
6  | 1 4 10 33 91 298 881 2825 8791 27947  87410 272991 ...
7  | 1 4 10 33 91 298 910 2974 9655 32287 108274 367489 ...
8  | 1 4 10 33 91 298 910 3017 9886 33767 116325 410298 ...
9  | 1 4 10 33 91 298 910 3017 9945 34124 118729 424498 ...
...

Andrew Howroyd, Table of n, a(n) for n = 1..1275
Andrew Howroyd, Additional PARI Programs

Rows 2..7 are A053307, A052365, A052366, A052367, A052372, A052373.
Main diagonal is A007716.
Cf. A214398, A246106, A304942, A318805.

permcount[v_List] := 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_List, q_List, 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!)];
Table[M[n-k+1, n-k+1, k], {n, 1, 12}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Sep 12 2018, after Andrew Howroyd *)

\\ see also link.
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)={1/prod(j=1, #q, (1-y^lcm(t,q[j]) + O(y*y^k))^gcd(t, q[j]))}
M(m, n, k)={my(s=0); forpart(q=m, s+=permcount(q)*polcoef(polcoef(exp(sum(t=1, n, K(q, t, k)/t*x^t) + O(x*x^n)), n), k)); s/m!}
for(n=1, 10, for(k=1, 12, print1(M(n, n, k), ", ")); print); \\ updated Andrew Howroyd, Mar 29 2020

T(n,k) = T(k,k) for n > k.

A052366 Number of nonnegative integer 4 X 4 matrices with sum of elements equal to n, under row and column permutations.

Original entry on oeis.org

1, 1, 4, 10, 33, 78, 224, 549, 1403, 3292, 7677, 16934, 36581, 75732, 152949, 298784, 569636, 1056500, 1916502, 3396630, 5901524, 10051384, 16820192, 27664756, 44795247, 71442327, 112366941, 174384376, 267289440, 404838044, 606375995

Offset: 0

Vladeta Jovovic, Mar 08 2000

nonn

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Row 4 of A318795.

Cf. A002724, A053307, A052365.

G.f.: (x^34 - 4*x^33 + 6*x^32 - x^31 + 3*x^30 - 11*x^29 + 33*x^28 - 19*x^27 + 81*x^26 - 52*x^25 + 152*x^24 - 36*x^23 + 255*x^22 - 130*x^21 + 367*x^20 - 84*x^19 + 350*x^18 - 94*x^17 + 350*x^16 - 84*x^15 + 367*x^14 - 130*x^13 + 255*x^12 - 36*x^11 + 152*x^10 - 52*x^9 + 81*x^8 - 19*x^7 + 33*x^6 - 11*x^5 + 3*x^4 - x^3 + 6*x^2 - 4*x + 1)/((x^10 + x^9 - x^7 - x^6 + x^4 + x^3 - x - 1)*(x^6 - 1)^2*(x^2 + 1)^3*(x^2 - 1)^4*(x^2 + x + 1)^2*(x + 1)*(x - 1)^9).

A054343 Number of nonnegative integer 3 X 3 matrices with sum of elements equal to n, under action of dihedral group of the square D_4.

Original entry on oeis.org

1, 3, 11, 31, 84, 198, 440, 904, 1766, 3266, 5802, 9906, 16384, 26284, 41104, 62752, 93831, 137589, 198309, 281249, 393148, 542154, 738480, 994320, 1324668, 1747220, 2283396, 2958228, 3801600, 4848120, 6138624, 7720032, 9647133, 11982423, 14798223, 18176499

Offset: 0

Vladeta Jovovic, May 05 2000

			There are 11 nonisomorphic nonnegative integer 3 X 3 matrices with sum of elements equal to 2, under action of D_4:
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 1] [0 0 0] [0 0 0] [0 0 0]
[0 0 0] [0 0 0] [0 0 1] [0 0 1] [0 1 0] [0 1 0] [1 0 1] [0 0 0] [0 0 0] [0 0 0] [0 2 0]
[0 1 1] [1 0 1] [0 1 0] [1 0 0] [0 0 1] [0 1 0] [0 0 0] [1 0 0] [0 0 2] [0 2 0] [0 0 0].

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-8,0,16,-24,16,8,-34,34,-8,-16,24,-16,0,8,-5,1).

Row n=3 of A343875.

Cf. A005232, A052365.

Vec((2*x^6+2*x^5+x^4+4*x^2-2*x+1)/((1-x^4)^2*(1-x^2)^2*(1-x)^5) + O(x^40)) \\ Colin Barker, Apr 26 2019

G.f.: (2*x^6+2*x^5+x^4+4*x^2-2*x+1)/((1-x^4)^2*(1-x^2)^2*(1-x)^5).

a(n) = 5*a(n-1) - 8*a(n-2) + 16*a(n-4) - 24*a(n-5) + 16*a(n-6) + 8*a(n-7) - 34*a(n-8) + 34*a(n-9) - 8*a(n-10) - 16*a(n-11) + 24*a(n-12) - 16*a(n-13) + 8*a(n-15) - 5*a(n-16) + a(n-17) for n>16. - Colin Barker, Apr 26 2019

A054975 Number of nonnegative integer 3 X 3 matrices with no zero rows or columns and with sum of elements equal to n, up to row and column permutation.

Original entry on oeis.org

1, 3, 13, 38, 97, 217, 453, 868, 1585, 2756, 4606, 7440, 11679, 17849, 26674, 39060, 56144, 79387, 110575, 151904, 206063, 276332, 366561, 481484, 626586, 808431, 1034636, 1314242, 1657500, 2076601, 2585262, 3199504, 3937370, 4819788

Offset: 3

Vladeta Jovovic, May 28 2000

			There are 3 nonnegative integer 3 X 3 matrices with no zero rows or columns and with sum of elements equal to 4, up to row and column permutation:
[0 0 1] [0 0 1] [0 0 1]
[0 0 1] [0 1 0] [0 1 0]
[1 1 0] [1 0 1] [2 0 0].

Andrew Howroyd, Table of n, a(n) for n = 3..1000

Column k=3 of A321615.

Cf. A052365.

gf := x^3*(x^14 - 2*x^13 + x^12 - 3*x^11 + 4*x^10 - 3*x^9 + 4*x^8 - x^7 - 4*x^6 + 2*x^5 - x^4 - 5*x^3 - 4*x^2 - 1)/((x^4 - x^3 + x - 1)*(x^3 - 1)^3*(x+1)^3*(x - 1)^5): s := series(gf, x, 101): for i from 3 to 100 do printf(`%d,`,coeff(s,x,i)) od:

G.f.: x^3*(x^14 - 2*x^13 + x^12 - 3*x^11 + 4*x^10 - 3*x^9 + 4*x^8 - x^7 - 4*x^6 + 2*x^5 - x^4 - 5*x^3 - 4*x^2 - 1)/((x^4 - x^3 + x - 1)*(x^3 - 1)^3*(x + 1)^3*(x - 1)^5).

More terms from James Sellers, May 29 2000

A052367 Number of nonnegative integer 5 X 5 matrices with sum of elements equal to n, under row and column permutations.

Original entry on oeis.org

1, 1, 4, 10, 33, 91, 277, 792, 2341, 6654, 18802, 51508, 138147, 359457, 910756, 2240915, 5365106, 12495406, 28353714, 62725603, 135469991, 285904968, 590347527, 1193817552, 2366907846, 4605225266, 8801576140, 16538061290

Offset: 0

Vladeta Jovovic, Mar 08 2000

nonn

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Row 5 of A318795.

Cf. A002724, A053307, A052365, A052366.

G.f.: - (x^86 - 3*x^85 + 9*x^84 + 12*x^83 + 59*x^82 + 116*x^81 + 452*x^80 + 736*x^79 + 2080*x^78 + 3344*x^77 + 7312*x^76 + 11708*x^75 + 21793*x^74 + 32869*x^73 + 55563*x^72 + 79389*x^71 + 123072*x^70 + 168321*x^69 + 243961*x^68 + 319938*x^67 + 438431*x^66 + 553731*x^65 + 724251*x^64 + 885383*x^63 + 1111989*x^62 + 1318149*x^61 + 1600579*x^60 + 1845557*x^59 +
2172889*x^58 + 2444070*x^57 + 2798839*x^56 + 3076865*x^55 + 3436180*x^54 + 3696058*x^53 + 4034590*x^52 + 4250683*x^51 + 4541020*x^50 + 4689359*x^49 + 4909073*x^48 + 4972196*x^47 + 5102026*x^46 + 5069013*x^45 + 5102464*x^44 + 4971700*x^43 + 4909948*x^42 + 4688757*x^41 + 4542211*x^40 + 4249809*x^39 + 4036170*x^38 + 3694857*x^37 + 3438025*x^36 +
3075494*x^35 + 2800760*x^34 + 2442552*x^33 + 2174743*x^32 + 1843864*x^31 + 1602482*x^30 + 1316113*x^29 + 1114023*x^28 + 883313*x^27 + 725930*x^26 + 551915*x^25 + 439662*x^24 + 318308*x^23 + 245205*x^22 + 166823*x^21 + 124009*x^20 + 78506*x^19 + 56071*x^18 + 32361*x^17 + 22208*x^16 + 11357*x^15 + 7673*x^14 + 3221*x^13 + 2294*x^12 + 684*x^11 + 594*x^10 + 59*x^9 + 133*x^8 + 21*x^7 + 18*x^6 - 2*x^4 - 3*x^3 + 9*x^2 - 5*x + 1) divided by (see next line)
((x^20 - 1)*(x^11 - x^10 + x^6 - x^5 + x - 1)*(x^7 - 2*x^6 + x^5 + x^4 - x^3 - x^2 + 2*x - 1)*(x^4 + x^3 + x^2 + x + 1)^4*(x^4 - x^3 + x^2 - x + 1)*(x^4 + x^2 + 1)*(x^2 + 1)^5*(x^2 + x + 1)^5*(x + 1)^11*(x - 1)^22).

A092091 Molien series for 9-dimensional group of structure Z_2 X Z_2 and order 4, corresponding to complete weight enumerators of Hermitian self-dual GF(3)-linear codes over GF(9).

Original entry on oeis.org

1, 4, 17, 52, 147, 360, 819, 1712, 3382, 6312, 11286, 19368, 32154, 51744, 81114, 124080, 185823, 272844, 393679, 558844, 781781, 1078792, 1470261, 1980576, 2639676, 3482960, 4553212, 5900496, 7584516, 9674496, 12252036, 15410976, 19260813, 23926548, 29552733

Offset: 0

N. J. A. Sloane, Apr 01 2004

Colin Barker, Table of n, a(n) for n = 0..1000
G. Nebe, E. M. Rains and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer, Berlin, 2006.
Index entries for Molien series
Index entries for linear recurrences with constant coefficients, signature (5,-6,-10,29,-9,-36,36,9,-29,10,6,-5,1).

Cf. A052365.

List([0..40], n-> ((315*(857 +167*(-1)^n) +60*(8347 +581*(-1)^n)*n + (384718 +6930*(-1)^n)*n^2 +84*(2027 +5*(-1)^n)*n^3 +48888*n^4 +9240*n^5 +1092*n^6 +72*n^7 +2*n^8))/322560 ); # G. C. Greubel, Feb 02 2020

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1-x+3*x^2+x^3)/((1+x)^4*(1-x)^9) )); // G. C. Greubel, Feb 02 2020

seq(coeff(series((1-x+3*x^2+x^3)/((1+x)^4*(1-x)^9), x, n+1), x, n), n = 0..40); # G. C. Greubel, Feb 02 2020

LinearRecurrence[{5,-6,-10,29,-9,-36,36,9,-29,10,6,-5,1}, {1,4,17,52,147,360, 819,1712,3382,6312,11286,19368,32154}, 35] (* Ray Chandler, Jul 15 2015 *)

Vec((1 -x +3*x^2 +x^3)/((1-x)^9*(1+x)^4) + O(x^40)) \\ Colin Barker, Jan 16 2017

def A092091_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1-x+3*x^2+x^3)/((1+x)^4*(1-x)^9) ).list()
A092091_list(40) # G. C. Greubel, Feb 02 2020

G.f.: (1 +2*x^2 +4*x^3 +x^4)/((1-x)^4*(1-x^2)^5).
G.f.: (1 -x +3*x^2 +x^3)/( (1+x)^4*(1-x)^9 ). - R. J. Mathar, Dec 18 2014
a(n) = ((315*(857+167*(-1)^n) + 60*(8347+581*(-1)^n)*n + (384718+6930*(-1)^n)*n^2 + 84*(2027+5*(-1)^n)*n^3 + 48888*n^4 + 9240*n^5 + 1092*n^6 + 72*n^7 + 2*n^8)) / 322560. - Colin Barker, Jan 16 2017

cp's OEIS Frontend

A007716 Number of polynomial symmetric functions of matrix of order n under separate row and column permutations.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

Extensions

A318795 Array read by antidiagonals: T(n,k) is the number of inequivalent nonnegative integer n X n matrices with sum of elements equal to k, under row and column permutations.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A052366 Number of nonnegative integer 4 X 4 matrices with sum of elements equal to n, under row and column permutations.

Views

Author

Keywords

Links

Crossrefs

Formula

A054343 Number of nonnegative integer 3 X 3 matrices with sum of elements equal to n, under action of dihedral group of the square D_4.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

Formula

A054975 Number of nonnegative integer 3 X 3 matrices with no zero rows or columns and with sum of elements equal to n, up to row and column permutation.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Formula

Extensions

A052367 Number of nonnegative integer 5 X 5 matrices with sum of elements equal to n, under row and column permutations.

Views

Author

Keywords

Links

Crossrefs

Formula

A092091 Molien series for 9-dimensional group of structure Z_2 X Z_2 and order 4, corresponding to complete weight enumerators of Hermitian self-dual GF(3)-linear codes over GF(9).

Views

Author

Keywords

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Sage

Formula