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A324564 Number T(n,k) of permutations p of [n] such that n-k is the maximum of 0 and the number of elements in any integer interval [p(i)..i+n*[i<p(i)]]; triangle T(n,k), n>=0, 0<=k<=n, read by rows. 13
1, 1, 0, 1, 1, 0, 4, 1, 1, 0, 15, 7, 1, 1, 0, 76, 31, 11, 1, 1, 0, 455, 185, 60, 18, 1, 1, 0, 3186, 1275, 435, 113, 29, 1, 1, 0, 25487, 10095, 3473, 1001, 215, 47, 1, 1, 0, 229384, 90109, 31315, 9289, 2299, 406, 76, 1, 1, 0, 2293839, 895169, 313227, 95747, 24610, 5320, 763, 123, 1, 1, 0 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,7
COMMENTS
Mirror image of A324563.
LINKS
Wikipedia, Iverson bracket
Wikipedia, Permutation
Wikipedia, Symmetric group
EXAMPLE
Triangle T(n,k) begins:
1;
1, 0;
1, 1, 0;
4, 1, 1, 0;
15, 7, 1, 1, 0;
76, 31, 11, 1, 1, 0;
455, 185, 60, 18, 1, 1, 0;
3186, 1275, 435, 113, 29, 1, 1, 0;
25487, 10095, 3473, 1001, 215, 47, 1, 1, 0;
...
Square array A(n,k) begins:
1, 0, 0, 0, 0, 0, ...
1, 1, 1, 1, 1, 1, ...
1, 1, 1, 1, 1, 1, ...
4, 7, 11, 18, 29, 47, ...
15, 31, 60, 113, 215, 406, ...
76, 185, 435, 1001, 2299, 5320, ...
455, 1275, 3473, 9289, 24610, 65209, ...
3186, 10095, 31315, 95747, 290203, 876865, ...
...
MAPLE
b:= proc(n, k) option remember; `if`(k>n, 0, `if`(k=0, n!,
LinearAlgebra[Permanent](Matrix(n, (i, j)->
`if`(j>=i and k+j<n+i or i>k+j, 1, 0)))))
end:
# as triangle:
T:= (n, k)-> b(n, k)-b(n, k+1):
seq(seq(T(n, k), k=0..n), n=0..10);
# as array:
A:= (n, k)-> b(n+k, k)-b(n+k, k+1):
seq(seq(A(d-k, k), k=0..d), d=0..10);
MATHEMATICA
b[n_, k_] := b[n, k] = If[k > n, 0, If[k == 0, n!, Permanent[Table[If[j >= i && k+j < n+i || i > k+j, 1, 0], {i, n}, {j, n}]]]];
(* as triangle: *)
T[n_, k_] := b[n, k] - b[n, k+1];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten
(* as array: *)
A[n_, k_] := b[n+k, k] - b[n+k, k+1];
Table[A[d-k, k], {d, 0, 10}, {k, 0, d}] // Flatten (* Jean-François Alcover, May 09 2019, after Alois P. Heinz *)
CROSSREFS
Columns k=0-10 give: A002467 (for n>0), A324621, A324622, A324623, A324624, A324625, A324626, A324627, A324628, A324629, A324630.
Diagonals of the triangle (rows of the array) n=0, (1+2), 3-10 give: A000007, A000012, A000032 (for n>=3), A324631, A324632, A324633, A324634, A324635, A324636, A324637.
Row sums give A000142.
T(2n,n) or A(n,n) gives A324638.
Sequence in context: A329637 A276834 A016684 * A276974 A122777 A103524
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Mar 06 2019
STATUS
approved

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Last modified July 22 09:18 EDT 2024. Contains 374485 sequences. (Running on oeis4.)