A327913 Array read by antidiagonals: T(n,m) is the number of distinct unordered row and column sums of n X m binary matrices.

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 7, 4, 1, 1, 5, 13, 13, 5, 1, 1, 6, 22, 34, 22, 6, 1, 1, 7, 34, 76, 76, 34, 7, 1, 1, 8, 50, 152, 221, 152, 50, 8, 1, 1, 9, 70, 280, 557, 557, 280, 70, 9, 1, 1, 10, 95, 482, 1264, 1736, 1264, 482, 95, 10, 1, 1, 11, 125, 787, 2630, 4766, 4766, 2630, 787, 125, 11, 1

Offset: 0

Author

Andrew Howroyd, Oct 30 2019

Keywords

• nonn
• tabl

Comments

Only matrices in which both row and columns sums are weakly increasing need to be considered. If order is also considered then the number of possibilities is given by A328887(n, m).

Examples

			Array begins:
=============================================
n\m | 0 1  2   3    4     5     6      7
----+----------------------------------------
  0 | 1 1  1   1    1     1     1      1 ...
  1 | 1 2  3   4    5     6     7      8 ...
  2 | 1 3  7  13   22    34    50     70 ...
  3 | 1 4 13  34   76   152   280    482 ...
  4 | 1 5 22  76  221   557  1264   2630 ...
  5 | 1 6 34 152  557  1736  4766  11812 ...
  6 | 1 7 50 280 1264  4766 15584  45356 ...
  7 | 1 8 70 482 2630 11812 45356 153228 ...
  ...
T(2,2) = 7. The following 7 matrices each have different row/column sums.
  [0 0]  [0 0]  [0 1]  [0 0]  [0 1]  [0 1]  [1 1]
  [0 0]  [0 1]  [1 0]  [1 1]  [0 1]  [1 1]  [1 1]
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..1325
• Manfred Krause, A simple proof of the Gale-Ryser theorem, American Mathematical Monthly, 1996.
• Wikipedia, Gale-Ryser theorem

Crossrefs

Main diagonal is A029894.

Cf. A028657 (nonequivalent binary n X m matrices).

Cf. A318396, A328887.

Programs

PARI

T(n,m)={local(Cache=Map());
  my(F(b, c, t, w)=my(hk=Vecsmall([b, c, t, w]), z);
     if(!mapisdefined(Cache, hk, &z),
       z = if(w&&c, sum(i=0, b, sum(j=ceil((t+i)/w), min(t+i, c), self()(i, j, t+i-j, w-1))), !t);
     mapput(Cache, hk, z)); z);
   F(n, n, 0, m)
}

Python

# After PARI implementation.
from functools import cache
@cache
def F(b, c, t, w):
    if w == 0:
        return 1 if t == 0 else 0
    return sum(
        sum(
            F(i, j, t + i - j, w - 1)
            for j in range((t + i - 1) // w, min(t + i, c) + 1)
        )
        for i in range(b + 1)
    )
A327913 = lambda n, m: F(n, n, 0, m)
for n in range(10):
    print([A327913(n, m) for m in range(0, 8)]) # Peter Luschny, Apr 09 2021