A058004 -id:A058004 - cp's OEIS frontend

A246106 Number A(n,k) of inequivalent n X n matrices with entries from [k], where equivalence means permutations of rows or columns; square array A(n,k), n>=0, k>=0, read by antidiagonals.

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 7, 1, 0, 1, 4, 27, 36, 1, 0, 1, 5, 76, 738, 317, 1, 0, 1, 6, 175, 8240, 90492, 5624, 1, 0, 1, 7, 351, 57675, 7880456, 64796982, 251610, 1, 0, 1, 8, 637, 289716, 270656150, 79846389608, 302752867740, 33642660, 1, 0

Offset: 0

Alois P. Heinz, Aug 13 2014

			Square array A(n,k) begins:
  1, 1,    1,        1,           1,              1, ...
  0, 1,    2,        3,           4,              5, ...
  0, 1,    7,       27,          76,            175, ...
  0, 1,   36,      738,        8240,          57675, ...
  0, 1,  317,    90492,     7880456,      270656150, ...
  0, 1, 5624, 64796982, 79846389608, 20834113243925, ...

Columns k = 0-10 give: A000007, A000012, A002724, A052269, A052271, A052272, A246112, A246113, A246114, A246115, A246116.
Rows n = 0-10 give: A000012, A001477, A039623, A058001, A058002, A058003, A058004, A246108, A246109, A246110, A246111.
Main diagonal gives A246107.
A028657, A242106, A353585 are related tables.
Cf. A242095, A256069.

b:= proc(n, i) option remember; `if`(n=0, [[]],
      `if`(i<1, [], [b(n, i-1)[], seq(map(p->[p[], [i, j]],
       b(n-i*j, i-1))[], j=1..n/i)]))
    end:
A:= proc(n, k) option remember; add(add(k^add(add(i[2]*j[2]*
      igcd(i[1], j[1]), j=t), i=s) /mul(i[1]^i[2]*i[2]!, i=s)
      /mul(i[1]^i[2]*i[2]!, i=t), t=b(n$2)), s=b(n$2))
    end:
seq(seq(A(n, d-n), n=0..d), d=0..10);

A246106(n,k)=A353585(k,n,n) \\ M. F. Hasler, May 01 2022

A(n,k) = Sum_{i=0..k} C(k,i) * A256069(n,i).

A(n,k) = Sum_{p,q in P(n)} k^Sum_{i in p, j in q} gcd(i, j) / (N(p)*N(q)) where N(p) = Product_{distinct parts x in p} x^m(x)*m(x)!, m(x) = multiplicity of x in p. - M. F. Hasler, Apr 30 2022 [corrected by Anders Kaseorg, Oct 04 2024]

A039623 a(n) = n^2*(n^2+3)/4.

Original entry on oeis.org

1, 7, 27, 76, 175, 351, 637, 1072, 1701, 2575, 3751, 5292, 7267, 9751, 12825, 16576, 21097, 26487, 32851, 40300, 48951, 58927, 70357, 83376, 98125, 114751, 133407, 154252, 177451, 203175, 231601, 262912, 297297, 334951, 376075, 420876, 469567

Offset: 1

Christian Meland (christian.meland(AT)pfi.no)

Previous definition was: Consider a figure like this <> (a squashed square, symmetric about both axes); each side is given 1 of n colors; a(n) = number of possibilities, allowing turning over.
Also number of 2 X 2 matrices with entries mod n, up to row and column permutation. 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. - Vladeta Jovovic, Nov 04 2000
Also, if a 2-set Y and a 3-set Z are disjoint subsets of an n-set X then a(n-5) is the number of 6-subsets of X intersecting both Y and Z. - Milan Janjic, Sep 08 2007

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Jean-Paul Delahaye, Le miraculeux "lemme de Burnside", pp. 145-6 in 'Pour la Science' (French edition of 'Scientific American'), No. 350, December 2006, Paris.
Milan Janjic, Two Enumerative Functions.
Milan Janjic and Boris Petkovic, A Counting Function, arXiv 1301.4550 [math.CO], 2013.
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A000217, A002724, A005353, A052271, A052272, A058001-A058004.

Row n=2 of A246106.

[n^2*(n^2+3)/4 : n in [1..50]]; // Wesley Ivan Hurt, Dec 26 2016

A039623:=n->n^2*(n^2+3)/4: seq(A039623(n), n=1..50); # Wesley Ivan Hurt, Dec 26 2016

Table[(n^2 (n^2+3))/4,{n,40}] (* or *) LinearRecurrence[{5,-10,10,-5,1},{1,7,27,76,175},40] (* Harvey P. Dale, Oct 01 2011 *)

Vec((-1-2*x-2*x^2-x^3)/(x-1)^5 + O(x^50)) \\ Michel Marcus, Aug 23 2015

a(n) = (1/4)*n^2*(n^2+3); \\ Altug Alkan, Apr 16 2016

From Harvey P. Dale, Oct 01 2011: (Start)
G.f.: (1 + 2*x + 2*x^2 + x^3)/(1 - x)^5.
a(1)=1, a(2)=7, a(3)=27, a(4)=76, a(5)=175; for n>5, a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5). (End)
E.g.f.: x*(4 + 10*x + 6*x^2 + x^3)*exp(x)/4. - Ilya Gutkovskiy, Apr 16 2016
a(n) = t(n-1)*t(n) + t(n-1) + t(n) where t=A000217. - J. M. Bergot, Apr 16 2016
a(n) = A000217(n)^2 - n*A000217(n-1). - Bruno Berselli, Feb 14 2017
a(n) = T(T(n-1)) + T(T(n)) where T(n) = A000217(n). - Charlie Marion, Feb 09 2023
Sum_{n>=1} 1/a(n) = 2*(1 + Pi^2 - sqrt(3)*Pi*coth(sqrt(3)*Pi))/9. - Amiram Eldar, Feb 13 2023
a(n) = binomial(n,2)*binomial(n+1,2) + n^2 = A006011(n) + A000290(n). - Detlef Meya, Nov 23 2023

More terms from Sam Alexander

Simplified the definition. - N. J. A. Sloane, Apr 20 2016

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

A058003 Number of 5 X 5 matrices with entries mod n, up to row and column permutation.

Original entry on oeis.org

1, 5624, 64796982, 79846389608, 20834113243925, 1979525296377132, 93242242505023122, 2625154125717590496, 49871029909245781491, 694584034909225304800, 7525039263469551291908, 66252712846754819753160

Offset: 1

Vladeta Jovovic, Nov 04 2000

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

Cf. A039623, A058001, A058002, A058004, A002724, A052271, A052272.

Row n=5 of A246106.

a(n)=(1/5!^2)*(n^25 + 20*n^20 + 100*n^17 + 70*n^15 + 300*n^14 + 225*n^13 + 400*n^12 + 400*n^11 + 100*n^10 + 1600*n^9 + 2300*n^8 + 1300*n^7 + 1200*n^6 + 1824*n^5 + 480*n^4 + 1680*n^3 + 2400*n^2).

More terms from James Sellers, Nov 08 2000

A058002 Number of 4 X 4 matrices with entries mod n, up to row and column permutation.

Original entry on oeis.org

1, 317, 90492, 7880456, 270656150, 4947097821, 58002778967, 490172624992, 3223155968811, 17382581357725, 79840867013666, 321169288917192, 1155731257886192, 3782368364610941, 11406226119319725, 32031530635953536, 84493500676300117, 210856844364222717

Offset: 1

Vladeta Jovovic, Nov 04 2000

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

Cf. A058001, A058003, A058004, A002724, A052271, A052272.

Row n=4 of A246106.

a(n)=(1/4!^2)*(n^16 + 12*n^12 + 36*n^10 + 67*n^8 + 160*n^6 + 204*n^4 + 96*n^2).

G.f.: -x*(x +1)*(x^14 +299*x^13 +84940*x^12 +6299584*x^11 +142482546*x^10 +1214416453*x^9 +4351647617*x^8 +6732281120*x^7 +4351647617*x^6 +1214416453*x^5 +142482546*x^4 +6299584*x^3 +84940*x^2 +299*x +1) / (x -1)^17. - Colin Barker, Jul 09 2013

More terms from Colin Barker, Jul 09 2013

cp's OEIS Frontend

A246106 Number A(n,k) of inequivalent n X n matrices with entries from [k], where equivalence means permutations of rows or columns; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

PARI

Formula

A039623 a(n) = n^2*(n^2+3)/4.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

PARI

Formula

Extensions

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A058003 Number of 5 X 5 matrices with entries mod n, up to row and column permutation.

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

Extensions

A058002 Number of 4 X 4 matrices with entries mod n, up to row and column permutation.

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

Extensions