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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.
1
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, 6894, 1267, 1, 1, 127, 4256, 46252, 162724, 150624, 10633, 1, 1, 255, 17509, 403316, 3231672, 8369812, 4900141, 105219, 1, 1, 511, 71349, 3415771, 59157822, 362855438, 696003275, 223813933, 1196889, 1
OFFSET
0,5
EXAMPLE
T(3,2) = 9:
The 3 X 3 Pascal matrix
[1 0 0]
[1 1 0]
[1 2 1]
has nine 2 X 2 submatrices
[1 0] [1 0] [0 0] [1 0] [1 0] [0 0] [1 1] [1 0] [1 0]
[1 1] [1 0] [1 0] [1 2] [1 1] [2 1] [1 2] [1 1] [2 1].
Sum of their permanents is 1 + 0 + 0 + 2 + 1 + 0 + 3 + 1 + 1 = 9.
Triangle T(n,k) begins:
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, 6894, 1267, 1;
1, 127, 4256, 46252, 162724, 150624, 10633, 1;
1, 255, 17509, 403316, 3231672, 8369812, 4900141, 105219, 1;
...
MAPLE
with(combinat): with(LinearAlgebra):
T:= proc(n, k) option remember; `if`(k=0 or k=n, 1, (l-> add(add(
Permanent(SubMatrix(Matrix(n, (i, j)-> binomial(i-1, j-1)),
i, j)), j in l), i in l))(choose([$1..n], k)))
end:
seq(seq(T(n, k), k=0..n), n=0..9);
MATHEMATICA
T[n_, k_] := T[n, k] = If[k == 0 || k == n, 1, Module[{l, M},
l = Subsets[Range[n], {k}];
M = Table[Binomial[i-1, j-1], {i, n}, {j, n}];
Total[Permanent /@ Flatten[Table[M[[i, j]], {i, l}, {j, l}], 1]]]];
Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Feb 29 2024 *)
CROSSREFS
Columns k=0-2 give: A000012, A000225, A369906.
Cf. A007318, A184173 (same for determinants).
Sequence in context: A176791 A259471 A220555 * A373398 A075440 A137470
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Jan 25 2024
STATUS
approved