OFFSET
0,16
COMMENTS
A singleton is a part of size 1. An endpoint is a vertex that appears in only one part.
T(n,k) is the number of binary matrices with n 1's, k rows and every row and column sum at least two up to permutation of rows and columns.
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..675 (rows 0..50)
FORMULA
T(2*n,n) = A307316(n).
EXAMPLE
Triangle begins:
1;
0;
0, 0;
0, 0;
0, 0, 1;
0, 0, 0;
0, 0, 1, 2;
0, 0, 0, 1;
0, 0, 1, 3, 5;
0, 0, 0, 3, 5;
0, 0, 1, 5, 17, 11;
0, 0, 0, 4, 20, 21;
0, 0, 1, 9, 53, 80, 34;
0, 0, 0, 6, 60, 167, 91;
0, 0, 1, 11, 121, 418, 410, 87;
0, 0, 0, 10, 149, 816, 1189, 402;
...
The T(4,2) = 1 partition is {{1,2},{1,2}}.
The corresponding matrix is:
[1 1]
[1 1]
The T(8,3) = 3 matrices are:
[1 1 1] [1 1 1 0] [1 1 1 1]
[1 1 1] [1 1 0 1] [1 1 0 0]
[1 1 0] [0 0 1 1] [0 0 1 1]
PROG
(PARI)
WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, (-1)^(n-1)/n))))-1, -#v)}
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}
K(q, t, k)={my(g=x*Ser(WeighT(Vec(sum(j=1, #q, my(g=gcd(t, q[j])); g*x^(q[j]/g)) + O(x*x^k), -k)))); (1-x)*g}
H(q, t, k)={my(c=sum(j=1, #q, if(t%q[j]==0, q[j]))); K(q, t, k) - c*x}
G(n, y=1)={my(s=0); forpart(q=n, s+=permcount(q)*exp(sum(t=1, n, subst(H(q, t, n\t)*y^t/t, x, x^t) ))); s/n!}
T(n)={my(v=Vec(G(n, 'y))); vector(#v, i, Vecrev(v[i], (i+1)\2))}
{ my(A=T(15)); for(i=1, #A, print(A[i])) }
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Andrew Howroyd, Feb 06 2024
STATUS
approved