A301272 - OEIS

%I #53 Sep 24 2023 10:30:40

%S 0,0,0,1,6,33,152,663,2778,11413,46332,186867,750878,3011025,12060480,

%T 48277423,193186146,772908429,3091983236,12368675691,49476275622,

%U 197908422985,791640682440,3166577409831,12666340397546,50665425902853,202661837829132,810647630936803

%N Number of derangements of S_n with exactly one peak.

%H Alois P. Heinz, <a href="/A301272/b301272.txt">Table of n, a(n) for n = 0..1663</a>

%H R. J. Cano, <a href="/A301272/a301272.txt">Sequencer program in PARI.</a>

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (6,-5,-16,12,16).

%F G.f.: x^3*(2*x^2+1)/((1-4*x)*(x+1)^2*(2*x-1)^2). - _Alois P. Heinz_, Apr 29 2018

%e a(3) = 1: 231.

%e a(4) = 6: 2143, 2341, 2413, 3142, 3412, 3421.

%p a:= n-> floor((<<0|1|0|0|0>, <0|0|1|0|0>, <0|0|0|1|0>, <0|0|0|0|1>,

%p         <16|12|-16|-5|6>>^n. <<1/8, 0, 0, 1, 6>>)[1, 1]):

%p seq(a(n), n=0..30);  # _Alois P. Heinz_, Apr 29 2018

%t Join[{0}, LinearRecurrence[{6, -5, -16, 12, 16}, {0, 0, 1, 6, 33}, 30]] (* _Jean-François Alcover_, May 31 2019 *)

%t CoefficientList[Series[x^3(2x^2+1)/((1-4x)(x+1)^2(2x-1)^2),{x,0,40}],x] (* _Harvey P. Dale_, Sep 01 2021 *)

%o (Python)

%o def count_peaks(pi):

%o     count = 0

%o     for i in range(i,len(pi)-1):

%o         if pi[i] > pi[i+1] and pi[i] > pi[i-1]:

%o             count += 1

%o     return count

%o def main(args):

%o     n = int(args[0])

%o     set = {1,2,...,n}

%o     drmts = []

%o     for pi in itertools.permutations(set):

%o         drmts.append(pi)

%o         for i in range(n):

%o             if pi[i] == i+1:

%o                 drmts.remove(pi)

%o                 break

%o     num = 0

%o     for pi in drmts:

%o         if count_peaks(pi) == 1:

%o             num += 1

%o     print('number of 1 peak derangements: ', num)

%o (PARI) A301272(n)={my(c=0,v,t,ok);for(k=0,n!-1,v=numtoperm(n,k);ok=1;for(i=1,n,if((v[i]==i),ok=0;break));if(ok,t=0;for(i=2,n-1,if((v[i]>v[i-1])&&(v[i]>v[i+1]),t++;if(t>1,break)));if(t==1,c++)));c} \\ _R. J. Cano_, Apr 25 2018

%o (PARI) \\ See Cano link.

%Y Cf. A000166, A001045, A216963.

%Y Column k=1 of A303564.

%K nonn,easy

%O 0,5

%A _Isabella Huang_, Mar 17 2018

%E a(10)-a(20) from _Alois P. Heinz_, Apr 25 2018

%E a(21)-a(27) from _Alois P. Heinz_, Apr 29 2018