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A369559 T(n,k) is the sum of the permanents of all k X k submatrices in the n X n Pascal matrix; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

%I A369559 #15 Feb 29 2024 06:58:06
%S A369559 1,1,1,1,3,1,1,7,9,1,1,15,50,35,1,1,31,234,482,185,1,1,63,1016,5011,
%T A369559 6894,1267,1,1,127,4256,46252,162724,150624,10633,1,1,255,17509,
%U A369559 403316,3231672,8369812,4900141,105219,1,1,511,71349,3415771,59157822,362855438,696003275,223813933,1196889,1
%N A369559 T(n,k) is the sum of the permanents of all k X k submatrices in the n X n Pascal matrix; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
%H A369559 Wikipedia, <a href="https://en.wikipedia.org/wiki/Permanent_(mathematics)">Permanent (mathematics)</a>
%e A369559 T(3,2) = 9:
%e A369559   The 3 X 3 Pascal matrix
%e A369559     [1 0 0]
%e A369559     [1 1 0]
%e A369559     [1 2 1]
%e A369559   has nine 2 X 2 submatrices
%e A369559     [1 0] [1 0] [0 0] [1 0] [1 0] [0 0] [1 1] [1 0] [1 0]
%e A369559     [1 1] [1 0] [1 0] [1 2] [1 1] [2 1] [1 2] [1 1] [2 1].
%e A369559   Sum of their permanents is 1 + 0 + 0 + 2 + 1 + 0 + 3 + 1 + 1 = 9.
%e A369559 Triangle T(n,k) begins:
%e A369559   1;
%e A369559   1,   1;
%e A369559   1,   3,     1;
%e A369559   1,   7,     9,      1;
%e A369559   1,  15,    50,     35,       1;
%e A369559   1,  31,   234,    482,     185,       1;
%e A369559   1,  63,  1016,   5011,    6894,    1267,       1;
%e A369559   1, 127,  4256,  46252,  162724,  150624,   10633,      1;
%e A369559   1, 255, 17509, 403316, 3231672, 8369812, 4900141, 105219, 1;
%e A369559   ...
%p A369559 with(combinat): with(LinearAlgebra):
%p A369559 T:= proc(n, k) option remember; `if`(k=0 or k=n, 1, (l-> add(add(
%p A369559       Permanent(SubMatrix(Matrix(n, (i, j)-> binomial(i-1, j-1)),
%p A369559        i, j)), j in l), i in l))(choose([$1..n], k)))
%p A369559     end:
%p A369559 seq(seq(T(n, k), k=0..n), n=0..9);
%t A369559 T[n_, k_] := T[n, k] = If[k == 0 || k == n, 1, Module[{l, M},
%t A369559     l = Subsets[Range[n], {k}];
%t A369559     M = Table[Binomial[i-1, j-1], {i, n}, {j, n}];
%t A369559     Total[Permanent /@ Flatten[Table[M[[i, j]], {i, l}, {j, l}], 1]]]];
%t A369559 Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Feb 29 2024 *)
%Y A369559 Columns k=0-2 give: A000012, A000225, A369906.
%Y A369559 Cf. A007318, A184173 (same for determinants).
%K A369559 nonn,tabl
%O A369559 0,5
%A A369559 _Alois P. Heinz_, Jan 25 2024