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A010028 Triangle read by rows: T(n,k) = 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. 9
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,5

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).

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.

FORMULA

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

EXAMPLE

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;

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 *)

CROSSREFS

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

Triangle in A086856 transposed. Cf. A001100.

Sequence in context: A086810 A085838 A094456 * A151860 A089627 A321686

Adjacent sequences:  A010025 A010026 A010027 * A010029 A010030 A010031

KEYWORD

tabl,nonn,changed

AUTHOR

N. J. A. Sloane.

STATUS

approved

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Last modified December 15 20:10 EST 2018. Contains 318154 sequences. (Running on oeis4.)