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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=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 Alois P. Heinz, Rows n = 0..23, flattened Wikipedia, Integer intervals Wikipedia, Iverson bracket Wikipedia, Permanent (mathematics) 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+jk+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. Cf. A002467, A324563. Sequence in context: A329637 A276834 A016684 * A276974 A122777 A103524 Adjacent sequences: A324561 A324562 A324563 * A324565 A324566 A324567 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.)