A318537 Irregular triangle read by rows: T(n,m) is the number of n X m (0,1)-matrices with pairwise distinct nonzero columns and pairwise distinct nonzero rows, n >= 0, m = 0..2^n-1.

1, 0, 1, 0, 0, 6, 6, 0, 0, 6, 174, 840, 2520, 5040, 5040, 0, 0, 0, 840, 24360, 335160, 3553200, 32382000, 259459200, 1816214400, 10897286400, 54486432000, 217945728000, 653837184000, 1307674368000, 1307674368000, 0, 0, 0, 2520, 335160, 15198120, 476496720, 12767000400, 314181504000, 7288444800000

Offset: 0

Author

Max Alekseyev, Aug 28 2018

Keywords

• nonn
• tabf

Comments

T(n,m) is divisible by both n! and m!, but not necessarily by n!*m!.

By symmetry T(n,m) = T(m,n).

T(n,2^n-1) = T(n,2^n-2) = (2^n-1)! = A028366(n).

Examples

			Triangle begins:
n=0: 1;
n=1: 0, 1;
n=2: 0, 0, 6, 6;
n=3: 0, 0, 6, 174, 840, 2520, 5040, 5040;
...
		

Crossrefs

Cf. A318538 (main diagonal), A059202.

Programs

PARI

{ A318537(n,m) = m! * sum(i=0,n, stirling(n+1,i+1)*binomial(2^i - 1,m)); }

Formula

T(n,m) = m! * Sum_{i=0..n} Stirling1(n+1,i+1) * binomial(2^i-1,m) = n! * Sum_{j=0..m} Stirling1(m+1,j+1) * binomial(2^j-1,n).

T(n,m) = A059202(n,m) * m!.