A321609 Array read by antidiagonals: T(n,k) is the number of inequivalent binary n X n matrices with k ones, under row and column permutations.

1, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 0, 3, 1, 1, 0, 0, 1, 3, 1, 1, 0, 0, 1, 6, 3, 1, 1, 0, 0, 0, 7, 6, 3, 1, 1, 0, 0, 0, 7, 16, 6, 3, 1, 1, 0, 0, 0, 6, 21, 16, 6, 3, 1, 1, 0, 0, 0, 3, 39, 34, 16, 6, 3, 1, 1, 0, 0, 0, 1, 44, 69, 34, 16, 6, 3, 1, 1, 0, 0, 0, 1, 55, 130, 90, 34, 16, 6, 3, 1, 1

Offset

0,13

Links

Table of n, a(n) for n=0..90.

Formula

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

T(n,k) = 0 for k > n^2.

Example

Array begins:
==========================================================
n\k| 0  1  2  3  4  5  6   7   8    9   10    11    12
---+------------------------------------------------------
0  | 1  0  0  0  0  0  0   0   0    0    0     0     0 ...
1  | 1  1  0  0  0  0  0   0   0    0    0     0     0 ...
2  | 1  1  3  1  1  0  0   0   0    0    0     0     0 ...
3  | 1  1  3  6  7  7  6   3   1    1    0     0     0 ...
4  | 1  1  3  6 16 21 39  44  55   44   39    21    16 ...
5  | 1  1  3  6 16 34 69 130 234  367  527   669   755 ...
6  | 1  1  3  6 16 34 90 182 425  870 1799  3323  5973 ...
7  | 1  1  3  6 16 34 90 211 515 1229 2960  6893 15753 ...
8  | 1  1  3  6 16 34 90 211 558 1371 3601  9209 24110 ...
9  | 1  1  3  6 16 34 90 211 558 1430 3825 10278 28427 ...
...

Mathematica

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[Product[Product[(1 + O[x]^(k + 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, n - k, k], {n, 0, 12}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Sep 10 2019, after Andrew Howroyd *)

Prog

(PARI)
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, k)={polcoef(prod(i=1, #p, prod(j=1, #q, (1 + x^lcm(p[i], q[j]) + O(x*x^k))^gcd(p[i], q[j]))), k)}
M(m, n, k)={my(s=0); forpart(p=m, forpart(q=n, s+=permcount(p) * permcount(q) * c(p, q, k))); s/(m!*n!)}
for(n=0, 10, for(k=0, 12, print1(M(n, n, k), ", ")); print); \\ Andrew Howroyd, Nov 14 2018

Crossrefs

Rows n=6..8 are A052370, A053304, A053305.

Main diagonal is A049311.

Row sums are A002724.

Cf. A052371 (as triangle), A057150, A246106, A318795.

Sequence in context: A204242 A211313 A368847 * A357638 A238414 A195151

Adjacent sequences: A321606 A321607 A321608 * A321610 A321611 A321612

Keyword

nonn,tabl

Author

Andrew Howroyd, Nov 14 2018

Status

approved