A129817 Number of alternating fixed-point-free permutations on n letters.

1, 0, 1, 1, 2, 6, 24, 102, 528, 2952, 19008, 131112, 1009728, 8271792, 74167488, 703077552, 7194754368, 77437418112, 890643066048, 10726837356672, 136988469649728, 1825110309733632, 25625477737660608, 374159217291201792, 5728724202727533888, 90961591766739121152, 1508303564683904357568, 25874345243221479539712, 461932949559928514787648, 8513674175717969079785472, 162818666826944872460200128

Offset: 0

Author

Vladeta Jovovic, May 20 2007

Keywords

• nonn

Comments

For n > 0, a(2n-1) = A129815(2n-1); for n > 1, a(2n) = A129815(2n) + A129815(2n-2). - Vladimir Shevelev, Apr 29 2008

We conjecture that for n >= 3, A000111(2n)/a(2n) < e < A000111(2n)/A129815(2n), so that A000111(2n)/a(2n) increases while A000111(2n)/A129815(2n) decreases (and both quotients tend to e). - Vladimir Shevelev, Apr 29 2008

From Emeric Deutsch, Aug 06 2009: (Start)

Alternating permutations are also called down-up permutations.

a(n) is also the number of alternating permutations of {1,2,...,n} having exactly 1 fixed point (see the Richard Stanley reference). Example: a(4)=2 because we have 4132 and 3241.

(End)

Examples

			a(4) = 2 because we have 3142 and 2143. - _Emeric Deutsch_, Aug 06 2009
		

Links

• R. P. Stanley, Alternating permutations and symmetric functions, arXiv:math/0603520 [math.CO], 2006.

Crossrefs

Cf. A000111, A000166, A007779.

Column k=0 of A162979.

Programs

Mathematica

nmax = 30;
fo = Exp[e*(ArcTan[q*t] - ArcTan[t])]/(1 - e*t);
fe = Sqrt[(1+t^2)/(1+q^2*t^2)]*Exp[e*(ArcTan[q*t] - ArcTan[t])]/(1-e*t);
Q[n_] := If [OddQ[n] ,  SeriesCoefficient[fo, {t, 0, n}],  SeriesCoefficient[fe, {t, 0, n}]] // Expand;
b[n_] :=  n!*SeriesCoefficient[Sec[x] + Tan[x], {x, 0, n}];
P[n_] := (Q[n] /. e^k_Integer :> b[k]) /. e :> b[1] // Expand;
a[n_] := Coefficient[P[n], q, 0];
Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 0, nmax}] (* Jean-François Alcover, Jul 24 2018 *)

Formula

a(n) = A162979(n,0). - Alois P. Heinz, Nov 24 2017

Extensions

a(21) from Alois P. Heinz, Nov 06 2015

a(0)=1 prepended by Alois P. Heinz, Nov 24 2017

a(22)..a(30) from Jean-François Alcover, Jul 24 2018