|
%I #32 Feb 02 2024 18:55:31
%S 1,0,1,1,2,6,24,102,528,2952,19008,131112,1009728,8271792,74167488,
%T 703077552,7194754368,77437418112,890643066048,10726837356672,
%U 136988469649728,1825110309733632,25625477737660608,374159217291201792,5728724202727533888,90961591766739121152,1508303564683904357568,25874345243221479539712,461932949559928514787648,8513674175717969079785472,162818666826944872460200128
%N Number of alternating fixed-point-free permutations on n letters.
%C For n > 0, a(2n-1) = A129815(2n-1); for n > 1, a(2n) = A129815(2n) + A129815(2n-2). - _Vladimir Shevelev_, Apr 29 2008
%C 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
%C From _Emeric Deutsch_, Aug 06 2009: (Start)
%C Alternating permutations are also called down-up permutations.
%C 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.
%C (End)
%H R. P. Stanley, <a href="https://arxiv.org/abs/math/0603520">Alternating permutations and symmetric functions</a>, arXiv:math/0603520 [math.CO], 2006.
%F a(n) = A162979(n,0). - _Alois P. Heinz_, Nov 24 2017
%e a(4) = 2 because we have 3142 and 2143. - _Emeric Deutsch_, Aug 06 2009
%t nmax = 30;
%t fo = Exp[e*(ArcTan[q*t] - ArcTan[t])]/(1 - e*t);
%t fe = Sqrt[(1+t^2)/(1+q^2*t^2)]*Exp[e*(ArcTan[q*t] - ArcTan[t])]/(1-e*t);
%t Q[n_] := If [OddQ[n] , SeriesCoefficient[fo, {t, 0, n}], SeriesCoefficient[fe, {t, 0, n}]] // Expand;
%t b[n_] := n!*SeriesCoefficient[Sec[x] + Tan[x], {x, 0, n}];
%t P[n_] := (Q[n] /. e^k_Integer :> b[k]) /. e :> b[1] // Expand;
%t a[n_] := Coefficient[P[n], q, 0];
%t Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 0, nmax}] (* _Jean-François Alcover_, Jul 24 2018 *)
%Y Cf. A000111, A000166, A007779.
%Y Column k=0 of A162979.
%K nonn
%O 0,5
%A _Vladeta Jovovic_, May 20 2007
%E a(21) from _Alois P. Heinz_, Nov 06 2015
%E a(0)=1 prepended by _Alois P. Heinz_, Nov 24 2017
%E a(22)..a(30) from _Jean-François Alcover_, Jul 24 2018
|
