A368401 Number T(n,k) of permutations of [n] whose sum of cycle maxima minus cycle minima gives k, triangle T(n,k), n>=0, 0<=k<=A002620(n), read by rows.

1, 1, 1, 1, 1, 2, 3, 1, 3, 7, 11, 2, 1, 4, 12, 28, 53, 12, 10, 1, 5, 18, 52, 135, 289, 84, 72, 58, 6, 1, 6, 25, 84, 257, 734, 1825, 524, 564, 496, 422, 60, 42, 1, 7, 33, 125, 429, 1407, 4545, 12983, 3520, 3976, 4292, 3950, 3422, 790, 486, 330, 24

Offset

0,6

Links

Alois P. Heinz, Rows n = 0..21, flattened

Wikipedia, Permutation

Formula

Sum_{k=0..A002620(n)} k * T(n,k) = A002538(n-1) for n >= 1.

Example

T(3,0) = 1: (1)(2)(3).
T(3,1) = 2: (12)(3), (1)(23).
T(3,2) = 3: (123), (132), (13)(2).
Triangle T(n,k) begins:
  1;
  1;
  1, 1;
  1, 2,  3;
  1, 3,  7, 11,   2;
  1, 4, 12, 28,  53,  12,   10;
  1, 5, 18, 52, 135, 289,   84,  72,  58,   6;
  1, 6, 25, 84, 257, 734, 1825, 524, 564, 496, 422, 60, 42;
  ...

Maple

b:= proc(n, s) option remember; `if`(n=0, 1, (k-> `if`(n>k,
      b(n-1, s)+add(b(n-1, subs(h=h+[0, 1], s)), h=s), 0)+
      `if`(n>k+1, b(n-1, {s[], [n, 1]}), 0)+add(h[2]!*expand(
      x^(h[1]-n)*b(n-1, s minus {h})), h=s))(nops(s)))
    end:
T:= (n, k)-> coeff(b(n, {}), x, k):
seq(seq(T(n, k), k=0..floor(n^2/4)), n=0..10);

Crossrefs

Row sums give A000142.

Cf. A002538, A002620, A124327, A143946, A367955, A368338.

Sequence in context: A062869 A102473 A011117 * A069269 A193092 A263484

Adjacent sequences: A368398 A368399 A368400 * A368402 A368403 A368404

Keyword

nonn,look,tabf

Author

Alois P. Heinz, Dec 22 2023

Status

approved