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A380392 Irregular triangle read by rows: T(n,k) is the number of n X n binary matrices containing k South-East paths of 1's connecting the top left and bottom right corners.

%I A380392 #16 Feb 22 2025 09:56:58
%S A380392 1,1,1,13,2,1,461,26,13,8,1,2,1,61708,1454,953,568,325,112,178,76,22,
%T A380392 46,48,2,16,4,4,8,8,0,1,2,1,32348492,340768,279142,168300,125121,
%U A380392 44436,81857,24666,25375,28182,19759,4476,17477,4334,7123,6436,4314,1708,5534
%N A380392 Irregular triangle read by rows: T(n,k) is the number of n X n binary matrices containing k South-East paths of 1's connecting the top left and bottom right corners.
%C A380392 A South-East path of 1's in a binary matrix is a path of connected 1's with steps South (0,-1) and East (1,0). Here 1's are said to be connected if they are adjacent in the same row or column.
%C A380392 Conjecture: The average number of South-East paths of 1's in all n X n binary matrices is A001790(n-1)/A101926(n-1). - _John Tyler Rascoe_, Feb 21 2025
%H A380392 John Tyler Rascoe, <a href="/A380392/b380392.txt">Rows n = 0..5, flattened</a>
%H A380392 John Tyler Rascoe, <a href="/A380392/a380392_1.py.txt">Python program</a>.
%e A380392 Triangle begins:
%e A380392      k=0   1   2  3  4  5  6
%e A380392  n=0   1;
%e A380392  n=1   1,  1;
%e A380392  n=2  13,  2,  1;
%e A380392  n=3 461, 26, 13, 8, 1, 2, 1;
%e A380392  ...
%e A380392 For row n = 3 the possible South-East paths are:
%e A380392  A.       B.       C.       D.       E.       F.
%e A380392  [1 1 1]  [1 1 0]  [1 1 0]  [1 0 0]  [1 0 0]  [1 0 0]
%e A380392  [0 0 1]  [0 1 1]  [0 1 0]  [1 1 1]  [1 1 0]  [1 0 0]
%e A380392  [0 0 1]  [0 0 1]  [0 1 1]  [0 0 1]  [0 1 1]  [1 1 1]
%e A380392 The 3 X 3 matrix below does not contain any of the paths A-F so it is counted under T(3,0) = 461.
%e A380392  [1 0 1]
%e A380392  [1 1 1]
%e A380392  [1 0 0]
%e A380392 The 3 X 3 matrix below contains paths A, B, and D so it is counted under T(3,3) = 8.
%e A380392  [1 1 1]
%e A380392  [1 1 1]
%e A380392  [1 0 1]
%o A380392 (Python) # see links
%Y A380392 Cf. A000984 (row lengths), A001790, A002416 (row sums), A086266, A101926, A261242, A369285.
%K A380392 nonn,tabf
%O A380392 0,4
%A A380392 _John Tyler Rascoe_, Jan 23 2025