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

0,5

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).

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

Example

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]

Prog

(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

Crossrefs

Main diagonal is A029894.

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

Cf. A318396, A328887.

Sequence in context: A094526 A088699 A101515 * A028657 A053534 A104881

Adjacent sequences: A327910 A327911 A327912 * A327914 A327915 A327916

Keyword

nonn,tabl

Author

Andrew Howroyd, Oct 30 2019

Status

approved