A001268 One-half the number of permutations of length n with exactly 4 rising or falling successions.

0, 0, 0, 0, 0, 1, 11, 113, 1099, 11060, 118484, 1366134, 16970322, 226574211, 3240161105, 49453685911, 802790789101, 13815657556958, 251309386257874, 4818622686395380, 97145520138758844, 2054507019515346789, 45484006970415223287, 1052036480881734378541

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Comments

(1/2) times number of permutations of 12...n such that exactly 4 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.
• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..200
• J. Riordan, A recurrence for permutations without rising or falling successions, Ann. Math. Statist. 36 (1965), 708-710.

Crossrefs

Cf. A002464, A000130, A086852. Equals A086855/2. A diagonal of A010028.

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:
a:= n-> ceil(coeff(S(n), t, 4)/2):
seq(a(n), n=0..25);  # Alois P. Heinz, Jan 11 2013

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]]]; a[n_] := Ceiling[Coefficient[S[n], t, 4]/2]; Table [a[n], {n, 0, 25}] (* Jean-François Alcover, Mar 24 2014, after Alois P. Heinz *)

Formula

Coefficient of t^4 in S[n](t) defined in A002464, divided by 2.

Recurrence (for n>5): (n-5)*(n^8 - 41*n^7 + 730*n^6 - 7358*n^5 + 45799*n^4 - 179702*n^3 + 432498*n^2 - 581244*n + 332100)*a(n) = (n^10 - 45*n^9 + 895*n^8 - 10301*n^7 + 75340*n^6 - 361190*n^5 + 1124682*n^4 - 2150033*n^3 + 2147364*n^2 - 499899*n - 544266)*a(n-1) - (n^10 - 44*n^9 + 869*n^8 - 10112*n^7 + 76390*n^6 - 388742*n^5 + 1336932*n^4 - 3028095*n^3 + 4237931*n^2 - 3198426*n + 917988)*a(n-2) - (n^10 - 43*n^9 + 823*n^8 - 9195*n^7 + 66108*n^6 - 318138*n^5 + 1033118*n^4 - 2224673*n^3 + 3023402*n^2 - 2325285*n + 761190)*a(n-3) + (n^8 - 33*n^7 + 471*n^6 - 3783*n^5 + 18594*n^4 - 56865*n^3 + 104723*n^2 - 104847*n + 42783)*(n-2)^2*a(n-4). - Vaclav Kotesovec, Aug 11 2013

a(n) ~ n!*exp(-2)/3. - Vaclav Kotesovec, Aug 11 2013