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

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

0,6

Links

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

Example

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).

Mathematica

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 *)

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)={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

Crossrefs

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

Row sums are A002724.

Cf. A049311.

Sequence in context: A124389 A366789 A027023 * A062278 A260638 A268523

Adjacent sequences: A052368 A052369 A052370 * A052372 A052373 A052374

Keyword

nonn,tabf

Author

Vladeta Jovovic, Mar 08 2000

Extensions

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

Status

approved