|
|
A071076
|
|
Number of permutations that avoid the generalized pattern 123-4.
|
|
4
|
|
|
1, 1, 2, 6, 23, 108, 598, 3815, 27532, 221708, 1970251, 19150132, 202064380, 2300071071, 28092017668, 366425723926, 5083645400819, 74745472084176, 1160974832572274, 18995175706664735, 326531476287842760, 5883736110875887560, 110893188848753125475
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,3
|
|
LINKS
|
|
|
FORMULA
|
E.g.f.: exp(int(A(y), y=0..x)), where A(y) = (sqrt(3)/2)*exp(y/2)/cos((sqrt(3)/2)*y + Pi/6).
Let b(n) = A049774(n) = number of permutations of [n] that avoid the consecutive pattern 123. Then a(n) = Sum_{i = 0..n-1} binomial(n-1,i)*b(i)*a(n-1-i) with a(0) = b(0) = 1. [See the recurrence for A_n and B_n in the proof of Theorem 13 in Kitaev's papers.] -
|
|
MAPLE
|
b:= proc(u, o, t) option remember; `if`(u+o=0, 1, add(
`if`(t=1 and o>j, 0, b(u+j-1, o-j, t+1)), j=1..o)+
add(b(u-j, o+j-1, 0), j=1..u))
end:
a:= n-> b(n, 0$2):
|
|
MATHEMATICA
|
b[u_, o_, t_] := b[u, o, t] = If[u + o == 0, 1, Sum[If[t == 1 && o > j, 0, b[u + j - 1, o - j, t + 1]], {j, 1, o}] + Sum[b[u - j, o + j - 1, 0], {j, 1, u}]]; a[n_] := b[n, 0, 0]; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Feb 01 2016, after Alois P. Heinz *)
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|