A242819 - OEIS

%I #36 Dec 01 2017 17:43:40

%S 1,1,2,6,21,3,90,30,450,270,2619,2322,99,17334,20772,2214,129114,

%T 195372,38394,1067661,1958337,591543,11259,9713682,20933154,8826246,

%U 443718,96393726,238789782,131367258,12450834,1036348587,2900868876,1989555210,297195804,3052323

%N Number T(n,k) of permutations of [n] with exactly k occurrences of the consecutive step pattern up, down, down; triangle T(n,k), n>=0, 0<=k<=max(0,floor((n-1)/3)), read by rows.

%C T(n,k) is also the number of permutations of [n] with exactly k occurrences of the consecutive step pattern up, up, down.

%C From _Vaclav Kotesovec_, Aug 26 2014: (Start)

%C Column k is asymptotic to c(k) * (3*sqrt(3)/(2*Pi))^n * n! * n^k.

%C Conjecture: c(k) = c(0) * (c(0)-1)^k / (3^k * k!).

%C Verified numerically:

%C c(0) = 1.96650951227123825842868... = (1+exp(Pi/sqrt(3)))*sqrt(3)/(2*Pi)

%C c(1) = 0.63355004986067503869384...

%C c(2) = 0.10205535828170995196503...

%C c(3) = 0.01095971939528021798...

%C c(4) = 0.000882722753946826148...

%C c(5) = 0.00005687732922585807984...

%C c(6) = 0.000003054026651631929902...

%C c(7) = 0.0000001405593242634352116...

%C c(8) = 0.00000000566049683079281633...

%C c(9) = 0.0000000002026268159682390665...

%C c(10)= 0.00000000000652802483581788974...

%C c(20)= 1.172921625090753...*10^(-28)

%C c(30)= 1.2959323...*10^(-47)

%C c(40)= 5.0751...*10^(-68)

%C (End)

%H Alois P. Heinz, <a href="/A242819/b242819.txt">Rows n = 0..120, flattened</a>

%e T(4,1) = 3: (1,4,3,2), (2,4,3,1), (3,4,2,1).

%e Triangle T(n,k) begins:

%e :  0 :       1;

%e :  1 :       1;

%e :  2 :       2;

%e :  3 :       6;

%e :  4 :      21,        3;

%e :  5 :      90,       30;

%e :  6 :     450,      270;

%e :  7 :    2619,     2322,      99;

%e :  8 :   17334,    20772,    2214;

%e :  9 :  129114,   195372,   38394;

%e : 10 : 1067661,  1958337,  591543,  11259;

%e : 11 : 9713682, 20933154, 8826246, 443718;

%p b:= proc(u, o, t) option remember; `if`(u+o=0, 1, expand(

%p       add(b(u-j, o+j-1, [1, 3, 1][t])*`if`(t=3, x, 1), j=1..u)+

%p       add(b(u+j-1, o-j, 2), j=1..o)))

%p     end:

%p T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 0, 1)):

%p seq(T(n), n=0..15);

%t b[u_, o_, t_] := b[u, o, t] = If[u+o == 0, 1, Expand[Sum[b[u-j, o+j-1, {1, 3, 1}[[t]]]*If[t == 3, x, 1], {j, 1, u}] + Sum[b[u+j-1, o-j, 2], {j, 1, o}]]]; T[n_] := Function[{p}, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, 0, 1]]; Table[T[n], {n, 0, 15}] // Flatten (* _Jean-François Alcover_, Feb 10 2015, after _Alois P. Heinz_ *)

%Y Columns k=0-10 give: A177479, A246246, A246247, A246248, A246249, A246250, A246251, A246252, A246253, A246254, A246255.

%Y Row sums give: A000142.

%Y Cf. A242783, A242784, A295987.

%K nonn,tabf

%O 0,3

%A _Alois P. Heinz_, May 23 2014