A007779 -id:A007779 - cp's OEIS frontend

A129815 Number of reverse alternating fixed-point-free permutations on n letters.

0, 0, 1, 2, 6, 22, 102, 506, 2952, 18502, 131112, 991226, 8271792, 73176262, 703077552, 7121578106, 77437418112, 883521487942, 10726837356672, 136104948161786, 1825110309733632

Offset: 1

Vladeta Jovovic, May 20 2007

From Emeric Deutsch, Aug 06 2009: (Start)
Reverse alternating permutations are called also up-down permutations.
a(n) is also the number of reverse alternating permutations having exactly 1 fixed point (see the Stanley reference). Example: a(4)=2 because we have 1423 and 2314.
(End)

			a(4)=2 because we have 3412 and 2413. [_Emeric Deutsch_, Aug 06 2009]

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

Cf. A000111, A000166, A007779, A129817.

a(2n-1) = A129817(2n-1). [Emeric Deutsch, Aug 06 2009]

a(21) from Alois P. Heinz, Jun 11 2015

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

Original entry on oeis.org

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

Vladeta Jovovic, May 20 2007

nonn

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)

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

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

Cf. A000111, A000166, A007779.

Column k=0 of A162979.

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 *)

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

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

A115455 a(n) = number of reverse alternating fixed-point-free involutions w on 1,2,...,2n, i.e., w(1) < w(2) > w(3) < w(4) > ... < w(2n), w^2=1 and w(i) != i for all i.

Original entry on oeis.org

1, 0, 1, 1, 4, 13, 59, 308, 1871, 12879, 99144, 843735, 7865177, 79698760, 872235089, 10253148625, 128839087676, 1723418002261, 24450430660739, 366702601116524, 5796979684239647, 96339860422218143, 1679159568980521104, 30628034488033962287

Offset: 0

Richard Stanley, Jan 22 2006

			a(3)=1 because there is one reverse alternating fixed-point-free involution on 1,...,6, viz., 351624.

Alois P. Heinz, Table of n, a(n) for n = 0..200
R. P. Stanley, Permutations, Joint Mathematics Meeting, 2009.
R. P. Stanley, Alternating permutations and symmetric functions, J. Comb. Theory A 114 (3) (2007) 436-460

Cf. A000085, A000111, A001147, A007779.

Table[SeriesCoefficient[(1-x^2)^(-1/4)*(1+x)^(-1/2)*Sum[(-1)^k*EulerE[2*k]*(1/4*Log[(1+x)/(1-x)])^k/k!,{k,0,n}],{x,0,n}],{n,0,20}] (* Vaclav Kotesovec, Apr 29 2014 *)

G.f.: (1-x^2)^{-1/4} (1+x)^{-1/2} Sum_{k>=0} E_{2k} v^k/k!, where E_{2k} is an Euler number and v = (1/4)*log((1+x)/(1-x)).

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A129815 Number of reverse alternating fixed-point-free permutations on n letters.

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

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A115455 a(n) = number of reverse alternating fixed-point-free involutions w on 1,2,...,2n, i.e., w(1) < w(2) > w(3) < w(4) > ... < w(2n), w^2=1 and w(i) != i for all i.

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