A092580 Triangle read by rows: T(n,k) is the number of permutations p of [n] in which exactly the first k terms satisfy the up-down property, i.e., p(1)<p(2), p(2)>p(3), p(3)<p(4), ...

1, 1, 1, 3, 1, 2, 12, 4, 3, 5, 60, 20, 15, 9, 16, 360, 120, 90, 54, 35, 61, 2520, 840, 630, 378, 245, 155, 272, 20160, 6720, 5040, 3024, 1960, 1240, 791, 1385, 181440, 60480, 45360, 27216, 17640, 11160, 7119, 4529, 7936, 1814400, 604800, 453600, 272160, 176400, 111600, 71190, 45290, 28839, 50521

Offset

1,4

Comments

Row sums are the factorial numbers (A000142). First column is A001710. Second column is A001715. Diagonal is A000111.

Links

Alois P. Heinz, Rows n = 1..150, flattened

E. Deutsch and W. P. Johnson, Create your own permutation statistics, Math. Mag., 77, 130-134, 2004.

Formula

T(n, k) = n!*[(k+1)*E(k)-E(k+1)]/(k+1)! for k<n and T(n, n) = E(n), where tan(x)+sec(x) = Sum_{n>=0} [E(n)x^n/n!] (i.e., E(n) = A000111(n)).

Sum_{k=0..n} (k+1) * T(n,k) = A230960(n). - Alois P. Heinz, Apr 27 2023

Example

T(4,3)=3 because 1432, 2431, 3421 are the only permutations of [4] in which exactly the first 3 entries satisfy the up-down property.
Triangle starts:
    1;
    1,   1;
    3,   1,  2;
   12,   4,  3,  5;
   60,  20, 15,  9, 16;
  360, 120, 90, 54, 35, 61;
  ...

Maple

b:= proc(u, o) option remember;
      `if`(u+o=0, 1, add(b(o-1+j, u-j), j=1..u))
    end:
E:= n-> b(n, 0):
T:= (n, k)-> `if`(n=k, E(n), n!*((k+1)*E(k)-E(k+1))/(k+1)!):
seq(seq(T(n, k), k=1..n), n=1..10);  # Alois P. Heinz, Aug 12 2016

Mathematica

b[u_, o_] := b[u, o] = If[u + o == 0, 1, Sum[b[o - 1 + j, u - j], {j, 1, u}]]; e[n_] := b[n, 0]; T[n_, k_] := If[n == k, e[n], n!*((k + 1)*e[k] - e[k + 1])/(k + 1)!]; Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* Jean-François Alcover, Dec 21 2016, after Alois P. Heinz *)

Crossrefs

Cf. A000111, A000142, A001710, A001715, A092582, A230960.

T(2n,n) gives A362603.

Sequence in context: A300930 A109528 A136125 * A004468 A254630 A145463

Adjacent sequences: A092577 A092578 A092579 * A092581 A092582 A092583

Keyword

nonn,tabl

Author

Emeric Deutsch and Warren P. Johnson (wjohnson(AT)bates.edu), Apr 10 2004

Status

approved