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 A324563 Number T(n,k) of permutations p of [n] such that 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. 20
 1, 0, 1, 0, 1, 1, 0, 1, 1, 4, 0, 1, 1, 7, 15, 0, 1, 1, 11, 31, 76, 0, 1, 1, 18, 60, 185, 455, 0, 1, 1, 29, 113, 435, 1275, 3186, 0, 1, 1, 47, 215, 1001, 3473, 10095, 25487, 0, 1, 1, 76, 406, 2299, 9289, 31315, 90109, 229384, 0, 1, 1, 123, 763, 5320, 24610, 95747, 313227, 895169, 2293839 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,10 COMMENTS Mirror image of triangle A324564. Or as array: Number A(n,k) of permutations p of [n+k] such that k is the maximum of 0 and the number of elements in any integer interval [p(i)..i+(n+k)*[i=0, k>=0, read by antidiagonals upwards. LINKS Alois P. Heinz, Rows n = 0..23, flattened Wikipedia, Integer intervals Wikipedia, Iverson bracket Wikipedia, Permanent (mathematics) Wikipedia, Permutation Wikipedia, Symmetric group FORMULA T(n,k) = |{ p in S_n : k = max_{i=1..n} (1+i-p(i)+n*[i0, T(0,0) = 1. T(n,k) = A008305(n,k) - A008305(n,k-1) for k > 0, T(n,0) = A000007(n). EXAMPLE T(4,1) = A(3,1) = 1: 1234. T(4,2) = A(2,2) = 1: 4123. T(4,3) = A(1,3) = 7: 1423, 1432, 3124, 3214, 3412, 4132, 4213. T(4,4) = A(0,4) = 15: 1243, 1324, 1342, 2134, 2143, 2314, 2341, 2413, 2431, 3142, 3241, 3421, 4231, 4312, 4321. Triangle T(n,k) begins: 1; 0, 1; 0, 1, 1; 0, 1, 1, 4; 0, 1, 1, 7, 15; 0, 1, 1, 11, 31, 76; 0, 1, 1, 18, 60, 185, 455; 0, 1, 1, 29, 113, 435, 1275, 3186; 0, 1, 1, 47, 215, 1001, 3473, 10095, 25487; ... Square array A(n,k) begins: 1, 1, 1, 4, 15, 76, 455, 3186, ... 0, 1, 1, 7, 31, 185, 1275, 10095, ... 0, 1, 1, 11, 60, 435, 3473, 31315, ... 0, 1, 1, 18, 113, 1001, 9289, 95747, ... 0, 1, 1, 29, 215, 2299, 24610, 290203, ... 0, 1, 1, 47, 406, 5320, 65209, 876865, ... ... MAPLE b:= proc(n, k) option remember; `if`(n=k, n!, LinearAlgebra[Permanent](Matrix(n, (i, j)-> `if`(i<=j and j b(n, k)-`if`(k=0, 0, b(n, k-1)): seq(seq(T(n, k), k=0..n), n=0..10); # as array: A:= (n, k)-> b(n+k, k)-`if`(k=0, 0, 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[n == k, n!, Permanent[Table[If[i <= j && j < k + i || n + j < k + i, 1, 0], {i, 1, n}, {j, 1, n}]]]; (* as triangle: *) T[n_, k_] := b[n, k] - If[k == 0, 0, b[n, k - 1]]; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* as array: *) A[n_, k_] := b[n + k, k] - If[k == 0, 0, b[n + k, k - 1]]; Table[A[d - k, k], {d, 0, 10}, {k, 0, d}] // Flatten (* Jean-François Alcover, May 08 2019, after Alois P. Heinz *) CROSSREFS Columns k=0, (1+2), 3-10 give: A000007, A000012, A000032 (for n>=3), A324631, A324632, A324633, A324634, A324635, A324636, A324637. Diagonals of the triangle (rows of the array) n=0-10 give: A002467 (for k>0), A324621, A324622, A324623, A324624, A324625, A324626, A324627, A324628, A324629, A324630. Row sums give A000142. T(2n,n) or A(n,n) gives A324638. Cf. A008305, A324564. Sequence in context: A096459 A293301 A218453 * A186372 A200893 A294583 Adjacent sequences: A324560 A324561 A324562 * A324564 A324565 A324566 KEYWORD nonn,tabl AUTHOR Alois P. Heinz, Mar 06 2019 STATUS approved

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Last modified March 30 10:57 EDT 2023. Contains 361609 sequences. (Running on oeis4.)