A010028 Triangle read by rows: T(n,k) is one-half the number of permutations of length n with exactly n-k rising or falling successions, for n >= 1, 1 <= k <= n. T(1,1) = 1 by convention.

1, 1, 0, 1, 2, 0, 1, 5, 5, 1, 1, 8, 24, 20, 7, 1, 11, 60, 128, 115, 45, 1, 14, 113, 444, 835, 790, 323, 1, 17, 183, 1099, 3599, 6423, 6217, 2621, 1, 20, 270, 2224, 11060, 32484, 56410, 55160, 23811, 1, 23, 374, 3950, 27152, 118484, 325322, 554306, 545135, 239653

Offset: 1

Author

N. J. A. Sloane

Keywords

• tabl
• nonn

Comments

(1/2) times number of permutations of 12...n such that exactly n-k of the following occur: 12, 23, ..., (n-1)n, 21, 32, ..., n(n-1).

Examples

			Triangle T(n,k) begins:
  1;
  1,  0;
  1,  2,   0;
  1,  5,   5,    1;
  1,  8,  24,   20,    7;
  1, 11,  60,  128,  115,   45;
  1, 14, 113,  444,  835,  790,  323;
  1, 17, 183, 1099, 3599, 6423, 6217, 2621;
  ...
		

References

• F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 263.

Links

• Alois P. Heinz, Rows n = 1..141, flattened
• J. Riordan, A recurrence for permutations without rising or falling successions, Ann. Math. Statist. 36 (1965), 708-710.

Crossrefs

Diagonals give A001266 (and A002464), A000130, A000349, A001267, A001268.

Triangle in A086856 transposed. Cf. A001100.

Row sums give A001710.

Programs

Maple

S:= proc(n) option remember; `if`(n<4, [1, 1, 2*t, 4*t+2*t^2]
       [n+1], expand((n+1-t)*S(n-1) -(1-t)*(n-2+3*t)*S(n-2)
       -(1-t)^2*(n-5+t)*S(n-3) +(1-t)^3*(n-3)*S(n-4)))
    end:
T:= (n, k)-> ceil(coeff(S(n), t, n-k)/2):
seq(seq(T(n, k), k=1..n), n=1..12);  # Alois P. Heinz, Dec 21 2012

Mathematica

S[n_] := S[n] = If[n<4, {1, 1, 2*t, 4*t+2*t^2}[[n+1]], Expand[(n+1-t)*S[n-1] - (1-t)*(n-2+3*t)*S[n-2]-(1-t)^2*(n-5+t)*S[n-3] + (1-t)^3*(n-3)*S[n-4]]]; T[n_, k_] := Ceiling[Coefficient[S[n], t, n-k]/2]; Table[Table[T[n, k], {k, 1, n}], {n, 1, 12}] // Flatten (* Jean-François Alcover, Jan 14 2014, translated from Alois P. Heinz's Maple code *)

Formula

For n>1, coefficient of t^(n-k) in S[n](t) defined in A002464, divided by 2.