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A242819
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.
14
1, 1, 2, 6, 21, 3, 90, 30, 450, 270, 2619, 2322, 99, 17334, 20772, 2214, 129114, 195372, 38394, 1067661, 1958337, 591543, 11259, 9713682, 20933154, 8826246, 443718, 96393726, 238789782, 131367258, 12450834, 1036348587, 2900868876, 1989555210, 297195804, 3052323
OFFSET
0,3
COMMENTS
T(n,k) is also the number of permutations of [n] with exactly k occurrences of the consecutive step pattern up, up, down.
From Vaclav Kotesovec, Aug 26 2014: (Start)
Column k is asymptotic to c(k) * (3*sqrt(3)/(2*Pi))^n * n! * n^k.
Conjecture: c(k) = c(0) * (c(0)-1)^k / (3^k * k!).
Verified numerically:
c(0) = 1.96650951227123825842868... = (1+exp(Pi/sqrt(3)))*sqrt(3)/(2*Pi)
c(1) = 0.63355004986067503869384...
c(2) = 0.10205535828170995196503...
c(3) = 0.01095971939528021798...
c(4) = 0.000882722753946826148...
c(5) = 0.00005687732922585807984...
c(6) = 0.000003054026651631929902...
c(7) = 0.0000001405593242634352116...
c(8) = 0.00000000566049683079281633...
c(9) = 0.0000000002026268159682390665...
c(10)= 0.00000000000652802483581788974...
c(20)= 1.172921625090753...*10^(-28)
c(30)= 1.2959323...*10^(-47)
c(40)= 5.0751...*10^(-68)
(End)
LINKS
EXAMPLE
T(4,1) = 3: (1,4,3,2), (2,4,3,1), (3,4,2,1).
Triangle T(n,k) begins:
: 0 : 1;
: 1 : 1;
: 2 : 2;
: 3 : 6;
: 4 : 21, 3;
: 5 : 90, 30;
: 6 : 450, 270;
: 7 : 2619, 2322, 99;
: 8 : 17334, 20772, 2214;
: 9 : 129114, 195372, 38394;
: 10 : 1067661, 1958337, 591543, 11259;
: 11 : 9713682, 20933154, 8826246, 443718;
MAPLE
b:= proc(u, o, t) option remember; `if`(u+o=0, 1, expand(
add(b(u-j, o+j-1, [1, 3, 1][t])*`if`(t=3, x, 1), j=1..u)+
add(b(u+j-1, o-j, 2), j=1..o)))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 0, 1)):
seq(T(n), n=0..15);
MATHEMATICA
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 *)
CROSSREFS
Row sums give: A000142.
Sequence in context: A074859 A162682 A103160 * A126099 A063753 A066893
KEYWORD
nonn,tabf
AUTHOR
Alois P. Heinz, May 23 2014
STATUS
approved