A302691 Number of broken alternating permutations of n things.

0, 0, 1, 2, 7, 26, 117, 594, 3407, 21682, 151853, 1160026, 9600567, 85566378, 817099909, 8322907298, 90074979487, 1032183177314, 12485056392285, 158964674218410, 2125201153260167, 29764791617545690, 435823661971532981, 6658895050949717362, 105979606291488794607

Offset

0,4

Comments

A permutation of {1,2,...,n} is said to be a "broken alternating permutation" if it is an alternating permutation (cf. A000111) except at one point. See El Hilany and Rau for precise definition and an explicit formula.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..484

D. Chebikin, Variations on descents and inversions in permutations, The Electronic J. of Combinatorics, 15 (2008), #R132.

Boulos El Hilany, Johannes Rau, Signed counts of real simple rational functions, arXiv:1712.05639 [math.AG], 2017, Proposition 6.4, p. 19.

Formula

a(n) ~ (4 - Pi) * 2^(n + 5/2) * n^(n + 3/2) / (exp(n) * Pi^(n + 3/2)). - Vaclav Kotesovec, Apr 14 2018

E.g.f.: (cos(x)-sin(x)+x-1)/(sin(x)-1). - Alois P. Heinz, Apr 16 2018

Maple

b:= proc(u, o, t) option remember;
      `if`(u+o=0, t, add(b(o+j-1, u-j, t), j=1..u)+
      `if`(t=0,      add(b(o-j, u-1+j, 1), j=1..o), 0))
    end:
a:= n-> b(n, 0$2):
seq(a(n), n=0..25);  # Alois P. Heinz, Apr 14 2018
# second Maple program:
egf:= (cos(x)-sin(x)+x-1)/(sin(x)-1):
a:= n-> n! * coeff(series(egf, x, n+1), x, n):
seq(a(n), n=0..25);  # Alois P. Heinz, Apr 16 2018

Mathematica

b[u_, o_, t_] := b[u, o, t] = If[u + o == 0, t, Sum[b[o + j - 1, u - j, t], {j, 1, u}] + If[t == 0, Sum[b[o - j, u - 1 + j, 1], {j, 1, o}], 0]];
a[n_] := b[n, 0, 0];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Apr 19 2018, after Alois P. Heinz *)

Crossrefs

Cf. A000111.

Column k=2 of A145876.

Sequence in context: A167551 A309396 A218670 * A081566 A213094 A141203

Adjacent sequences: A302688 A302689 A302690 * A302692 A302693 A302694

Keyword

nonn

Author

Michael De Vlieger, Apr 11 2018

Extensions

a(13)-a(24) from Alois P. Heinz, Apr 14 2018

Status

approved