A002724 -id:A002724 - cp's OEIS frontend

A052365 Number of nonnegative integer 3 X 3 matrices with sum of elements equal to n, under row and column permutations.

1, 1, 4, 10, 24, 51, 114, 219, 424, 768, 1352, 2278, 3759, 5978, 9328, 14181, 21164, 30943, 44560, 63063, 88088, 121321, 165152, 222157, 295857, 389948, 509456, 659697, 847552, 1080452, 1367814, 1719652, 2148596, 2668107, 3294676, 4046069

Offset: 0

Vladeta Jovovic, Mar 08 2000

nonn

Also Molien series for group of structure S_3 X S_3 = (Z_3 X Z_3).O_2^+(3) and order 36, corresponding to complete weight enumerators of Hermitian self-dual GF(3)-linear codes over GF(9) containing the all-ones vector.

Andrew Howroyd, 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

Row 3 of A318795.

Cf. A002724, A053307, A052366, A052267, A092091.

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!)];
a[n_] := M[3, 3, n];
a /@ Range[0, 40] (* Jean-François Alcover, Sep 03 2019, after Andrew Howroyd in A318795 *)

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

Another form for g.f.: u1/u2, where u1 := 1 + x + 2*x^3 + 10*x^4 + 17*x^5 + 19*x^6 + 20*x^7 + 29*x^8 + 37*x^9 + 34*x^10 + 23*x^11 + 12*x^12 + 7*x^13 + 3*x^14 + x^15 u2 := (1-x^2)^4*(1-x^3)^4*(1-x^6);

A053304 Number of 7 X 7 binary matrices with n=0..49 ones up to row and column permutations.

Original entry on oeis.org

1, 1, 3, 6, 16, 34, 90, 211, 515, 1229, 2960, 6893, 15753, 34450, 72235, 143477, 269186, 473945, 781713, 1203617, 1728192, 2310376, 2874232, 3325215, 3576980, 3576980, 3325215, 2874232, 2310376, 1728192, 1203617, 781713, 473945, 269186

Offset: 0

Vladeta Jovovic, Mar 05 2000

Andrew Howroyd, Table of n, a(n) for n = 0..49

Row 7 of A052371 and A321609.

Cf. A002724, A049311.

\\ See A321609 for M.
vector(50, n, M(7,7,n-1))

a(n) = A049311(n) for n <= 7.

Sum_{n=0..49} a(n) = 33642660 = A002724(7).

A122082 Number of unlabeled bicolored graphs on 2n nodes which are invariant when the two color classes are interchanged.

Original entry on oeis.org

1, 2, 5, 16, 67, 404, 3904, 64840, 1930842, 104698904, 10401039400, 1900637187280, 641429385018832, 401454435464761376, 467919402404052870944, 1019758699013228238271040, 4171161230867751509749228304

Offset: 0

N. J. A. Sloane, Oct 18 2006

nonn

R. W. Robinson, Numerical implementation of graph counting algorithms, AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1976.

Andrew Howroyd, Table of n, a(n) for n = 0..50
F. Harary, L. March and R. W. Robinson, On enumerating certain design problems in terms of bicolored graphs with no isolates, Environment and Planning, B 5 (1978), 31-43.
F. Harary, L. March and R. W. Robinson, On enumerating certain design problems in terms of bicolored graphs with no isolates, Environment and Planning B: Urban Analytics and City Science, 5 (1978), 31-43. [Annotated scanned copy]

Row sums of A123548.

Cf. A002724, A007139, A333160.

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];
edges[v_] := Sum[GCD[v[[i]], v[[j]]], {i, 2, Length[v]}, {j, 1, i - 1}] + Total @ Quotient[v + 1, 2]
a[n_] := (s=0; Do[s += permcount[p]*2^edges[p], {p, IntegerPartitions[n]}]; s/n!);
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Jul 06 2018, after Andrew Howroyd *)

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}
edges(v) = {sum(i=2, #v, sum(j=1, i-1, gcd(v[i],v[j]))) + sum(i=1, #v, (v[i]+1)\2)}
a(n) = {my(s=0); forpart(p=n, s+=permcount(p)*2^edges(p)); s/n!} \\ Andrew Howroyd, Oct 23 2017

a(n) = 2*A007139(n) - A002724(n). - Vladeta Jovovic, Feb 27 2007

More terms from Vladeta Jovovic, Feb 27 2007

A002725 Number of incidence matrices: n X (n+1) binary matrices under row and column permutations.

Original entry on oeis.org

1, 3, 13, 87, 1053, 28576, 2141733, 508147108, 402135275365, 1073376057490373, 9700385489355970183, 298434346895322960005291, 31479360095907908092817694945, 11474377948948020660089085281068730, 14568098446466140788730090352230460100956

Offset: 0

N. J. A. Sloane

a(0) = 1 by convention.

			a(1) = 3: [0,0], [0,1], [1,1].
a(2) = 13:
000 000 000 000 001 001 001 001 001 011 011 011 111
000 001 011 111 001 010 011 110 111 011 101 111 111

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Andrew Howroyd, Table of n, a(n) for n = 0..50 (terms 0..23 from Alois P. Heinz)
A. Kerber, Experimentelle Mathematik, Séminaire Lotharingien de Combinatoire. Institut de Recherche Math. Avancée, Université Louis Pasteur, Strasbourg, Actes 19 (1988), 77-83. [Annotated scanned copy]
B. Misek, On the number of classes of strongly equivalent incidence matrices, (Czech with English summary) Casopis Pest. Mat. 89 1964 211-218.

Cf. A002623, A002727, A006148, A002728, A002724.

A diagonal of the array A(m,n) described in A028657. - N. J. A. Sloane, Sep 01 2013

b:= proc(n, i) option remember; `if`(n=0, {0}, `if`(i<1, {},
      {seq(map(p-> p+j*x^i, b(n-i*j, i-1))[], j=0..n/i)}))
    end:
a:= n-> add(add(2^add(add(igcd(i, j)* coeff(s, x, i)*
      coeff(t, x, j), j=1..degree(t)), i=1..degree(s))/
      mul(i^coeff(s, x, i)*coeff(s, x, i)!, i=1..degree(s))/
      mul(i^coeff(t, x, i)*coeff(t, x, i)!, i=1..degree(t)),
      t=b(n+1$2)), s=b(n$2)):
seq(a(n), n=0..12);  # Alois P. Heinz, Aug 01 2014

b[n_, i_] := b[n, i] = If[n == 0, {0}, If[i<1, {}, Flatten @ Table[ Map[ Function[ {p}, p+j*x^i], b[n-i*j, i-1]], {j, 0, n/i}]]];
a[n_] := Sum[Sum[2^Sum[ Sum [ GCD[i, j]*Coefficient[s, x, i]*Coefficient[t, x, j], {j, 1, Exponent[t, x]}], {i, 1, Exponent[s, x]}] / Product[ i^Coefficient[s, x, i]*Coefficient[s, x, i]!, {i, 1, Exponent[s, x]}] / Product[i^Coefficient[t, x, i]*Coefficient[t, x, i]!, {i, 1, Exponent[t, x]}], {t, b[n+1, n+1]}], {s,  b[n, n]}];
Table[a[n], {n, 0, 12}] (* Jean-François Alcover, Feb 25 2015, after Alois P. Heinz *)

a(n) = A(n+1,n) \\ A defined in A028657. - Andrew Howroyd, Mar 01 2023

a(n) = sum_{1*s_1+2*s_2+...=n, 1*t_1+2*t_2+...=n+1} (fix A[s_1, s_2, ...; t_1, t_2, ...]/(1^s_1*s_1!*2^s_2*s_2!*...*1^t_1*t_1!*2^t_2*t_2!*...)) where fix A[...] = 2^sum_{i, j>=1} (gcd(i, j)*s_i*t_j). - Sean A. Irvine, Jul 31 2014

More terms from Vladeta Jovovic, Feb 04 2000

A006383 Number of equivalence classes of n X n binary matrices when one can permute rows, permute columns and complement columns.

Original entry on oeis.org

1, 1, 3, 7, 41, 299, 6128, 343656, 67013431, 45770163273, 108577103160005, 886929528971819040, 24943191706060101926577, 2425246700258693990625775794, 820270898724825121532156178527106

Offset: 0

N. J. A. Sloane

			a(2) = 3:
00 10 11
00 00 00

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Andrew Howroyd, Table of n, a(n) for n = 0..50 (terms 0..35 from Sean A. Irvine)
M. A. Harrison, On the number of classes of binary matrices, IEEE Trans. Computers, 22 (1973), 1048-1051.
M. A. Harrison, On the number of classes of binary matrices, IEEE Transactions on Computers, C-22.12 (1973), 1048-1052. (Annotated scanned copy)
Index entries for sequences related to binary matrices

Main diagonal of A363349.

Cf. A002724, A005232, A006381, A006382, A056204, A056205.

Definition corrected by Brendan McKay, Jan 07 2007

Terms a(7) onward from Max Alekseyev, Feb 05 2010

A052269 Number of n X n matrices over GF(3) up to row and column permutations.

Original entry on oeis.org

1, 3, 27, 738, 90492, 64796982, 302752867740, 9610448114487414, 2130536585704570302966, 3379836486315342147630795474, 39197947672609240635681299333726499, 3385559039111928075792568062997302563515455, 2212558055097091715366351569353345370930731329332056

Offset: 0

Vladeta Jovovic, Feb 04 2000

nonn

Cf. A002724, A052271, A052272, A091060.

Column k=3 of A246106.

A052269(n)=A353585(3,n,n) \\ M. F. Hasler, Apr 30 2022

a(n) = Sum_{1*s_1+2*s_2+...=n, 1*t_1+2*t_2+...=n} (fixA[s_1, s_2, ...;t_1, t_2, ...]/(1^s_1*s_1!*2^s_2*s_2!*...*1^t_1*t_1!*2^t_2*t_2!*...)) where fixA[...] = 3^Sum_{i, j>=1} (gcd(i,j)*s_i*t_j). - Christian G. Bower, Dec 18 2003

More terms from Alois P. Heinz, Jul 31 2014

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).

A052371 Triangle T(n,k) of n X n binary matrices with k=0...n^2 ones up to row and column permutations.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 3, 6, 7, 7, 6, 3, 1, 1, 1, 1, 3, 6, 16, 21, 39, 44, 55, 44, 39, 21, 16, 6, 3, 1, 1, 1, 1, 3, 6, 16, 34, 69, 130, 234, 367, 527, 669, 755, 755, 669, 527, 367, 234, 130, 69, 34, 16, 6, 3, 1, 1

Offset: 0

Vladeta Jovovic, Mar 08 2000

			Triangle begins:
  1;
  1, 1;
  1, 1, 3, 1, 1;
  1, 1, 3, 6, 7, 7, 6, 3, 1, 1;
  1, 1, 3, 6, 16, 21, 39, 44, 55, 44, 39, 21, 16, 6, 3, 1, 1;
  ...
(the last block giving the numbers of 4 X 4 binary matrices with k=0..16 ones up to row and column permutations).

Andrew Howroyd, Table of n, a(n) for n = 0..2890 (rows n=0..20)

Rows 6..8 are A052370, A053304, A053305.
Row sums are A002724.
Cf. A049311.

permcount[v_] := Module[{m = 1, s = 0, t, i, k = 0}, 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_] := Product[(1 + x^LCM[p[[i]], q[[j]]])^GCD[p[[i]], q[[j]]], {i, 1, Length[p]}, {j, 1, Length[q]}];
row[n_] := Module[{s = 0}, Do[Do[s += permcount[p]*permcount[q]*c[p, q], {q, IntegerPartitions[n]}], {p, IntegerPartitions[n]}]; CoefficientList[ s/(n!^2), x]]
row /@ Range[0, 5] // Flatten (* Jean-François Alcover, Sep 22 2019, after Andrew Howroyd *)

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}
c(p, q)={prod(i=1, #p, prod(j=1, #q, (1 + x^lcm(p[i], q[j]))^gcd(p[i], q[j])))}
row(n)={my(s=0); forpart(p=n, forpart(q=n, s+=permcount(p) * permcount(q) * c(p, q))); Vec(s/(n!^2))}
for(n=1, 5, print(row(n))) \\ Andrew Howroyd, Nov 14 2018

a(0)=1 prepended by Andrew Howroyd, Nov 14 2018

A058001 Number of 3 X 3 matrices with entries mod n, up to row and column permutation.

Original entry on oeis.org

1, 36, 738, 8240, 57675, 289716, 1144836, 3780288, 10865205, 27969700, 65834406, 143887536, 295467263, 575308020, 1069960200, 1911933696, 3298486761, 5516122788, 8972008810, 14233690800, 22078652211, 33555443636, 50058302988, 73417387200, 106006948125

Offset: 1

Vladeta Jovovic, Nov 04 2000

Number of k X l matrices with entries mod n, up to row and column permutation is Z(S_k X S_l; n,n,...) where Z(S_k X S_l; x_1,x_2,...) is cycle index of Cartesian product of symmetric groups S_k and S_l of degree k and l, respectively.

Alois P. Heinz, Table of n, a(n) for n = 1..1000
Marko R. Riedel, Number of equivalence classes of matrices, Math Stackexchange.
Marko R. Riedel, Computing the cycle index for arbitary k x l matrices using Maple
Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).

Cf. A058002, A058003, A058004, A002724, A052271, A052272.

Row n=3 of A246106.

CoefficientList[Series[x (12x^7+369x^6+2514x^5+4375x^4+2360x^3+423x^2+26x+1)/(x-1)^10,{x,0,30}],x] (* or *) LinearRecurrence[{10,-45,120,-210,252,-210,120,-45,10,-1},{0,1,36,738,8240,57675,289716,1144836,3780288,10865205},30] (* Harvey P. Dale, Nov 23 2024 *)

a(n) = (1/3!^2)*(n^9 + 6*n^6 + 9*n^5 + 8*n^3 + 12*n^2).

G.f.: x*(12*x^7+369*x^6+2514*x^5+4375*x^4+2360*x^3+423*x^2+26*x+1) / (x-1)^10. - Colin Barker, Jul 09 2013

A058004 Number of 6 X 6 matrices with entries mod n, up to row and column permutation.

Original entry on oeis.org

1, 251610, 302752867740, 9178323524804624, 28125393244553141210, 19909522361922032493690, 5116530046996205504668323, 626072069382507442113224128, 43460016875695276108491159279

Offset: 1

Vladeta Jovovic, Nov 04 2000

Number of k X l matrices with entries mod n, up to row and column permutation is Z(S_k X S_l; n,n,...) where Z(S_k X S_l; x_1,x_2,...) is cycle index of Cartesian product of symmetric groups S_k and S_l of degree k and l, respectively.

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Cf. A039623, A058001-A058003, A002724, A052271, A052272.

Row n=6 of A246106.

a(n)=(1/6!^2)*(n^36 + 30*n^30 + 225*n^26 + 170*n^24 + 1350*n^22 + 3225*n^20 + 4075*n^18 + 9900*n^16 + 28500*n^14 + 56048*n^12 + 61020*n^10 + 77616*n^8 + 153840*n^6 + 87840*n^4 + 34560*n^2).

cp's OEIS Frontend

A052365 Number of nonnegative integer 3 X 3 matrices with sum of elements equal to n, under row and column permutations.

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A053304 Number of 7 X 7 binary matrices with n=0..49 ones up to row and column permutations.

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PARI

Formula

A122082 Number of unlabeled bicolored graphs on 2n nodes which are invariant when the two color classes are interchanged.

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Mathematica

PARI

Formula

Extensions

A002725 Number of incidence matrices: n X (n+1) binary matrices under row and column permutations.

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Maple

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A006383 Number of equivalence classes of n X n binary matrices when one can permute rows, permute columns and complement columns.

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Extensions

A052269 Number of n X n matrices over GF(3) up to row and column permutations.

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PARI

Formula

Extensions

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

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Keywords

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Formula

A052371 Triangle T(n,k) of n X n binary matrices with k=0...n^2 ones up to row and column permutations.

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