A091061 -id:A091061 - cp's OEIS frontend

A242095 Number A(n,k) of inequivalent n X n matrices with entries from [k], where equivalence means permutations of rows or columns or the symbol set; square array A(n,k), n>=0, k>=0, read by antidiagonals.

1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 5, 1, 0, 1, 1, 8, 18, 1, 0, 1, 1, 9, 139, 173, 1, 0, 1, 1, 9, 408, 15412, 2812, 1, 0, 1, 1, 9, 649, 332034, 10805764, 126446, 1, 0, 1, 1, 9, 749, 2283123, 3327329224, 50459685390, 16821330, 1, 0

Offset: 0

Alois P. Heinz, Aug 14 2014

A(n,k) = A(n,k+1) for k >= n^2.

			A(2,2) = 5:
  [1 1]  [2 1]  [2 2]  [2 1]  [2 1]
  [1 1], [1 1], [1 1], [2 1], [1 2].
Square array A(n,k) begins:
  1, 1,    1,        1,          1,            1, ...
  0, 1,    1,        1,          1,            1, ...
  0, 1,    5,        8,          9,            9, ...
  0, 1,   18,      139,        408,          649, ...
  0, 1,  173,    15412,     332034,      2283123, ...
  0, 1, 2812, 10805764, 3327329224, 173636442196, ...

Columns k=0-10 give: A000007, A000012, A091059, A091060, A091061, A091062, A246122, A246123, A246124, A246125, A246126.
Main diagonal gives A091058.
A(n,n^2) gives A091057.
Cf. A242093, A242106, A246106.

with(numtheory):
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:= proc(n, k) option remember; add(add(add(mul(mul(add(d*
      coeff(u, x, d), d=divisors(ilcm(i, j)))^(igcd(i, j)*
      coeff(s, x, i)*coeff(t, x, j)), j=1..degree(t)),
      i=1..degree(s))/mul(i^coeff(u, x, i)*coeff(u, x, i)!,
      i=1..degree(u))/mul(i^coeff(t, x, i)*coeff(t, x, i)!,
      i=1..degree(t))/mul(i^coeff(s, x, i)*coeff(s, x, i)!,
      i=1..degree(s)), u=b(k$2)), t=b(n$2)), s=b(n$2))
    end:
seq(seq(A(n, d-n), n=0..d), d=0..10);

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_, k_] := A[n, k] = Sum[Sum[Sum[Product[Product[With[{g = GCD[i, j]* Coefficient[s, x, i]*Coefficient[t, x, j]}, If[g == 0, 1, Sum[d* Coefficient[u, x, d], {d, Divisors[LCM[i, j]]}]^g]], {j, Exponent[t, x]} ],
{i, Exponent[s, x]}]/Product[i^Coefficient[u, x, i]*Coefficient[u, x, i]!,
{i, Exponent[u, x]}]/Product[i^Coefficient[t, x, i]*Coefficient[t, x, i]!,
{i, Exponent[t, x]}]/Product[i^Coefficient[s, x, i]*Coefficient[s, x, i]!,
{i, Exponent[s, x]}], {u, b[k, k]}], {t, b[n, n]}], {s, b[n, n]}];
Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 10}] // Flatten (* Jean-François Alcover, Feb 21 2016, after Alois P. Heinz, updated Jan 01 2021 *)

A052271 Number of n X n matrices over GF(4) under row and column permutations.

Original entry on oeis.org

1, 4, 76, 8240, 7880456, 79846389608, 9178323524804624, 12508419942924578958856, 209493560585995285291677153144, 44407122853769773657258254744483639216, 122042291850117110186411151373496788803833567344, 4435666701292795500500326090033525002278314431436080593856

Offset: 0

Vladeta Jovovic, Feb 05 2000

nonn

Cf. A002724, A052269, A052272, A091061.

Column k=4 of A246106.

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[...] = 4^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

A091062 Number of n X n matrices over symbol set {1,2,3,4,5} equivalent under any permutation of row, columns or the symbol set.

Original entry on oeis.org

1, 1, 9, 649, 2283123, 173636442196, 234378355489344704, 5830719097591168695360621, 2779203181367458204944451774688032, 26174539685600184383643311230836752183522328, 4992259182572292655057303928366260085844535079288641049

Offset: 0

Christian G. Bower, Dec 17 2003

nonn

Cf. A091057-A091061.

Column k=5 of A242095.

a(n) = sum_{1*s_1+2*s_2+...=n, 1*t_1+2*t_2+...=n, 1*u_1+2*u_2+...=5} (fixA[s_1, s_2, ...;t_1, t_2, ...;u_1, u_2, ...]/(1^s_1*s_1!*2^s_2*s_2!* ... *1^t_1*t_1!*2^t_2*t_2!* ... *1^u_1*u_1!*2^u_2*u_2!*...)) where fixA[...] = prod_{i, j>=1} ( (sum_{d|lcm(i, j)} (d*u_d))^(gcd(i, j)*s_i*t_j)).

a(10) from Alois P. Heinz, Aug 14 2014

A091058 Number of n X n matrices over symbol set {1,...,n} equivalent under any permutation of row, columns or the symbol set.

Original entry on oeis.org

1, 1, 5, 139, 332034, 173636442196, 27652322898323351716, 2006943506669869627232430791792, 95763314593596534914617136274432901605313744, 4114852471732264714685900791520508800628430894815984377778000

Offset: 0

Christian G. Bower, Dec 17 2003

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..14
C. G. Bower, Explanation of A091057-A091062
Index to number of inequivalent matrices modulo permutation of rows and columns

Cf. A091057, A091059, A091060, A091061, A091062.

Main diagonal of A242095.

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}]]];
A242095[n_, k_] := A242095[n, k] = With[{co = Coefficient, ex = Exponent}, Sum[Sum[Sum[Product[Product[With[{g = GCD[i, j]*co[s, x, i]*co[t, x, j]}, If[g == 0, 1, Sum[d*co[u, x, d], {d, Divisors[LCM[i, j]]}]^g]], {j, ex[t, x]}], {i, ex[s, x]}]/Product[i^co[u, x, i]*co[u, x, i]!, {i, ex[u, x]}]/Product[i^co[t, x, i]*co[t, x, i]!, {i, ex[t, x]}]/Product[i^co[s, x, i]*co[s, x, i]!, {i, ex[s, x]}], {u, b[k, k]}], {t, b[n, n]}], {s, b[n, n]}]];
a[n_] := A242095[n, n];
Table[Print[n, " ", a[n]]; a[n], {n, 0, 12}] (* Jean-François Alcover, May 29 2023, after Alois P. Heinz in A242095 *)

Pol. = InfinitePolynomialRing(QQ)
@cached_function
def Z(n):
    if n == 0: return Pol.one()
    return sum(x[k]*Z(n-k) for k in (1..n))/n
@cached_function
def monprod(M):
    p = Pol.one()
    V = [m.variables() for m in M]
    T = cartesian_product(V)
    for t in T:
        r = [Pol.varname_key(str(u))[1] for u in t]
        j = [Pol(M[u[0]]).degree(u[1]) for u in enumerate(t)]
        lcm_r = lcm(r)
        p *= x[lcm_r]^(prod(r)/lcm_r*prod(j))
    return p
@cached_function
def pol_isotop(n,k):
    P = Z(n)
    p = Pol.zero()
    coeffs = P.coefficients()
    mons = P.monomials()
    C = cartesian_product(k*[mons])
    Csorted = [tuple(sorted(u)) for u in C]
    Cset = set(Csorted)
    for c in Cset:
        p += Csorted.count(c)*prod([coeffs[mons.index(u)] for u in c])*monprod(c)
    return p
@cached_function
def rule_sub(r,m):
    D = 0
    for d in divisors(r):
        try: D += d*m.degrees()[-d-1]
        except: break
    return D
def a(n,k=2):
    P = Z(n)
    coeffs = P.coefficients()
    Q = pol_isotop(n,k)
    inds = [Pol.varname_key(str(u))[1] for u in Q.variables()]
    p = 0
    for mon in enumerate(P.monomials()):
        m = Pol(mon[1])
        p += coeffs[mon[0]]*Q.subs({x[i]:rule_sub(i,m) for i in inds})
    return p
# Philip Turecek, Jun 17 2023

a(n) = Sum_{1*s_1+2*s_2+...=n, 1*t_1+2*t_2+...=n, 1*u_1+2*u_2+...=n} (fixA[s_1, s_2, ...;t_1, t_2, ...;u_1, u_2, ...]/ (1^s_1*s_1!*2^s_2*s_2!*... *1^t_1*t_1!*2^t_2*t_2!*... *1^u_1*u_1!*2^u_2*u_2!*...)) where fixA[...] = Product_{i, j>=1} ( (Sum_{d|lcm(i, j)} (d*u_d))^(gcd(i, j)*s_i*t_j)).

a(n) asymptotic to n^(n^2)/(n!^3) = A002489(n)/(A000142(n)^3).

a(9) from Alois P. Heinz, Aug 14 2014

A353585 Square array T(n,k): row n lists the number of inequivalent matrices over Z/nZ, modulo permutations of rows and columns, of size r X c, 1 <= r <= c, c >= 1.

Original entry on oeis.org

1, 1, 2, 1, 3, 3, 1, 7, 6, 4, 1, 4, 27, 10, 5, 1, 13, 10, 76, 15, 6, 1, 36, 92, 20, 175, 21, 7, 1, 5, 738, 430, 35, 351, 28, 8, 1, 22, 15, 8240, 1505, 56, 637, 36, 9, 1, 87, 267, 35, 57675, 4291, 84, 1072, 45, 10, 1, 317, 5053, 1996, 70, 289716, 10528, 120, 1701, 55, 11

Offset: 1

M. F. Hasler, Apr 28 2022

The array is read by falling antidiagonals.
Each row lists the number of inequivalent matrices of size 1 X 1, then 2 X 1, 2 X 2, then 3 X 1, 3 X 2, 3 X 3, etc., with coefficients in Z/nZ (or equivalently, in {1, ..., n}). See Examples for more.
Row 1 counts the zero matrices, there is only one of any size. Row 2 counts binary matrices, this is the lower triangular part of A028657, without the trivial row & column 0. (This table might have been extended with a trivial column 0 = A000012 (counting the 1 matrix of size 0) and row 0 = A000007 counting the number of r X c matrices with no entry, as done in A246106.)
The square matrices (size 1 X 1, 2 X 2, 3 X 3, ...) are counted in columns with triangular numbers, k = T(r) = r(r+1)/2 = (1, 3, 6, 10, 15, ...) = A000217.

			The table starts
   n \ k=1,  2,   3,   4,   5,   6, ...: T(n,k)
  ----+--------------------------------------
   1  |  1   1    1    1    1     1 ...
   2  |  2   3    7    4   13    36 ...
   3  |  3   6   27   10   92   738 ...
   4  |  4  10   76   20  430  8240 ...
   5  |  5  15  175   35 1505 57675 ...
  ...
Columns 2, 3 and 4, 5, 6 correspond to matrices of size 1 X 2, 2 X 2 and 1 X 3, 2 X 3, 3 X 3, respectively.
Column 4 says that there are (1, 4, 10, 20, 35, ...) inequivalent matrices of size 1 X 3 with entries in Z/nZ (n = 1, 2, 3, 4, ...); these numbers are given by (n+2 choose 3) = binomial(n+2, 3) = n(n+1)(n+2)/6 = A000292(n).

OEIS Index to number of inequivalent matrices modulo permutation of row and columns.

All of the following related sequences can be expressed in terms of T(n, k, r) := T(n, k(k-1)/2 + r), WLOG r <= k:
A028657(n,k) = A353585(2,n,k): inequivalent m X n binary matrices,
A002723(n) = T(2,n,2): size n X 2, A002724(n) = T(2,n,n): size n X n,
A002727(n) = T(2,n,3): size n X 3, A002725(n) = T(2,n,n+1): size n X (n+1),
A006148(n) = T(2,n,4): size n X 4, A002728(n) = T(2,n,n+2): size n X (n+2),
A052264(n) = T(2,n,5): size n X 5,
A052269(n) = T(3,n,n): number of inequivalent ternary matrices of size n X n,
A052271(n) = T(4,n,n): number of inequivalent matrices over Z/4Z of size n X n,
A052272(n) = T(5,n,n): number of inequivalent matrices over Z/5Z of size n X n,
A246106(n,k) = A353585(k,n,n): number of inequivalent n X n matrices over Z/kZ, and its diagonal A091058 and columns 1, 2, ..., 10: A000012, A091059, A091060, A091061, A091062, A246122, A246123, A246124, A246125, A246126.

A353585(n,k,r)={if(!r,r=sqrtint(8*k)\/2; k-=r*(r-1)\2); my(m(c, p=1, L=0)=for(i=1,#c, if(i==#c || c[i+1]!=c[i], p *= c[i]^(i-L)*(i-L)!; L=i )); p, S=0); forpart(P=k, my(T=0); forpart(Q=r, T += n^sum(i=1,#P, sum(j=1,#Q, gcd(P[i],Q[j]) ))/m(Q)); S += T/m(P)); S}

Let k = c(c-1)/2 + r, 1 <= r <= c, then
T(n, c, r) := T(n, k) = Sum_{p in P(c), q in P(r)} n^S(p, q)/(N(p)*N(q)), where P(r) are the partitions of r, S(p, q) = Sum_{i in p, j in q} gcd(i, j), N(p) = Product_{distinct parts x in p} x^m(x)*m(x)!, m(x) = multiplicity of x in p.
(See, e.g., A080577 for a list of partitions of positive integers.)
In particular:
T(n, 1) = n, T(n, 2) = n(n+1)/2 = A000217(n), T(n, 4) = C(n+2, 3) = A000292(n), T(n, 7) = C(n+3, 4) = A000332(n+3), etc.: T(n, k(k+1)/2 + 1) = C(n+k, k+1),
T(n, k(k+1)/2) = A246106(k, n).

cp's OEIS Frontend

A242095 Number A(n,k) of inequivalent n X n matrices with entries from [k], where equivalence means permutations of rows or columns or the symbol set; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A052271 Number of n X n matrices over GF(4) under row and column permutations.

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A091062 Number of n X n matrices over symbol set {1,2,3,4,5} equivalent under any permutation of row, columns or the symbol set.

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A091058 Number of n X n matrices over symbol set {1,...,n} equivalent under any permutation of row, columns or the symbol set.

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A353585 Square array T(n,k): row n lists the number of inequivalent matrices over Z/nZ, modulo permutations of rows and columns, of size r X c, 1 <= r <= c, c >= 1.

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