|
|
A264173
|
|
Number T(n,k) of permutations of [n] with exactly k (possibly overlapping) occurrences of the consecutive pattern 1324; triangle T(n,k), n>=0, 0<=k<=max(0,floor(n/2-1)), read by rows.
|
|
14
|
|
|
1, 1, 2, 6, 23, 1, 110, 10, 632, 86, 2, 4229, 782, 29, 32337, 7571, 407, 5, 278204, 78726, 5856, 94, 2659223, 882997, 84351, 2215, 14, 27959880, 10657118, 1251246, 48234, 322, 320706444, 137977980, 19318314, 984498, 14322, 42, 3985116699, 1910131680, 311306106
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,3
|
|
COMMENTS
|
Pattern 4231 gives the same triangle.
|
|
LINKS
|
|
|
FORMULA
|
Sum_{k>0} k * T(n,k) = ceiling((n-3)*n!/4!) = A061206(n-3) (for n>3).
|
|
EXAMPLE
|
T(4,1) = 1: 1324.
T(6,2) = 2: 132546, 142536.
T(8,3) = 5: 13254768, 13264758, 14253768, 14263758, 15263748.
T(10,4) = 14: 132547698(10), 132548697(10), 132647598(10), 132648597(10), 132748596(10), 142537698(10), 142538697(10), 142637598(10), 142638597(10), 142738596(10), 152637498(10), 152638497(10), 152738496(10), 162738495(10).
Triangle T(n,k) begins:
00 : 1;
01 : 1;
02 : 2;
03 : 6;
04 : 23, 1;
05 : 110, 10;
06 : 632, 86, 2;
07 : 4229, 782, 29;
08 : 32337, 7571, 407, 5;
09 : 278204, 78726, 5856, 94;
10 : 2659223, 882997, 84351, 2215, 14;
|
|
MAPLE
|
b:= proc(u, o, t) option remember; expand(`if`(u+o=0, 1,
add(b(u-j, o+j-1, `if`(t>0 and j<t, -j, 0)), j=1..u)+
add(b(u+j-1, o-j, j)*`if`(t<0 and -j<=t, x, 1), j=1..o)))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 0$2)):
seq(T(n), n=0..14);
|
|
MATHEMATICA
|
b[u_, o_, t_] := b[u, o, t] = Expand[If[u + o == 0, 1, Sum[b[u - j, o + j - 1, If[t > 0 && j < t, -j, 0]], {j, 1, u}] + Sum[b[u + j - 1, o - j, j] * If[t < 0 && -j <= t, x, 1], {j, 1, o}]]];
T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][ b[n, 0, 0]];
|
|
CROSSREFS
|
Columns k=0-10 give: A113228, A264174, A264175, A264176, A264177, A264178, A264179, A264180, A264181, A264182, A264183.
T(2n+2,n) gives A000108(n) for n>0.
|
|
KEYWORD
|
nonn,tabf
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|