A000130 -id:A000130 - cp's OEIS frontend

A086855 Number of permutations of length n with exactly 4 rising or falling successions.

0, 0, 0, 0, 0, 2, 22, 226, 2198, 22120, 236968, 2732268, 33940644, 453148422, 6480322210, 98907371822, 1605581578202, 27631315113916, 502618772515748, 9637245372790760, 194291040277517688, 4109014039030693578, 90968013940830446574, 2104072961763468757082

Offset: 0

N. J. A. Sloane, Aug 19 2003

nonn

Permutations of 12...n such that exactly 4 of the following occur: 12, 23, ..., (n-1)n, 21, 32, ..., n(n-1).

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

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.

Cf. A002464, A000130, A000349, A001267, A086852, A086853. A diagonal of A001100.

Twice A001268.

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)):
seq(a(n), n=0..25);  # Alois P. Heinz, Jan 11 2013

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]]; Table [a[n], {n, 0, 25}] (* Jean-François Alcover, Oct 13 2014, after Alois P. Heinz *)

Coefficient of t^4 in S[n](t) defined in A002464.

a(n) ~ 2/3*exp(-2) * n!. - Vaclav Kotesovec, Aug 14 2013

A383857 Number of permutations of [n] such that precisely one rising or falling succession occurs, but without either n(n-1) or (n-1)n.

Original entry on oeis.org

0, 0, 2, 8, 34, 196, 1366, 10928, 98330, 983036, 10811134, 129714184, 1686103522, 23603603540, 354033474374, 5664286296416, 96289603698346, 1733166940314028, 32929480177913230, 658578501071986616, 13829959293448920434, 304255691156335505924

Offset: 1

Wolfdieter Lang, May 19 2025

See A086852 or 2*A000130 for the counting including the successions n(n-1) and (n-1)n. See also the k = 1 columns of the triangles A001100 and 2*A086856.

For the number of permutations of length n without rising or falling successions see A002464(n).

			a(3) = 2*1 from the permutations 213 and the reverted 312.
a(4) = 2*4 from 1324, 1423, 2314, 3124 and the reverted 4231, 3241, 4132, 4213.
a(5) = 2*17 from the permutations corresponding to A086852(5) = 2*20, without 13542, 24513, 25413, and the reverted 24531, 31542, 31452.

Cf. A000130, A001100, A002464, A086852, A086856.

a(n) = A002464(n+1) - (n-1) * A002464(n).

cp's OEIS Frontend

A086855 Number of permutations of length n with exactly 4 rising or falling successions.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Maple

Mathematica

Formula