This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A353585 #20 May 14 2023 14:35:42
%S A353585 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,
%T A353585 1,5,738,430,35,351,28,8,1,22,15,8240,1505,56,637,36,9,1,87,267,35,
%U A353585 57675,4291,84,1072,45,10,1,317,5053,1996,70,289716,10528,120,1701,55,11
%N 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.
%C A353585 The array is read by falling antidiagonals.
%C A353585 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.
%C A353585 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.)
%C A353585 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.
%H A353585 <a href="/index/Mat#inequiv">OEIS Index to number of inequivalent matrices modulo permutation of row and columns</a>.
%F A353585 Let k = c(c-1)/2 + r, 1 <= r <= c, then
%F A353585 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.
%F A353585 (See, e.g., A080577 for a list of partitions of positive integers.)
%F A353585 In particular:
%F A353585 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),
%F A353585 T(n, k(k+1)/2) = A246106(k, n).
%e A353585 The table starts
%e A353585 n \ k=1, 2, 3, 4, 5, 6, ...: T(n,k)
%e A353585 ----+--------------------------------------
%e A353585 1 | 1 1 1 1 1 1 ...
%e A353585 2 | 2 3 7 4 13 36 ...
%e A353585 3 | 3 6 27 10 92 738 ...
%e A353585 4 | 4 10 76 20 430 8240 ...
%e A353585 5 | 5 15 175 35 1505 57675 ...
%e A353585 ...
%e A353585 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.
%e A353585 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).
%o A353585 (PARI) 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}
%Y A353585 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:
%Y A353585 A028657(n,k) = A353585(2,n,k): inequivalent m X n binary matrices,
%Y A353585 A002723(n) = T(2,n,2): size n X 2, A002724(n) = T(2,n,n): size n X n,
%Y A353585 A002727(n) = T(2,n,3): size n X 3, A002725(n) = T(2,n,n+1): size n X (n+1),
%Y A353585 A006148(n) = T(2,n,4): size n X 4, A002728(n) = T(2,n,n+2): size n X (n+2),
%Y A353585 A052264(n) = T(2,n,5): size n X 5,
%Y A353585 A052269(n) = T(3,n,n): number of inequivalent ternary matrices of size n X n,
%Y A353585 A052271(n) = T(4,n,n): number of inequivalent matrices over Z/4Z of size n X n,
%Y A353585 A052272(n) = T(5,n,n): number of inequivalent matrices over Z/5Z of size n X n,
%Y A353585 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.
%K A353585 nonn,tabl
%O A353585 1,3
%A A353585 _M. F. Hasler_, Apr 28 2022