|
| |
|
|
A271206
|
|
Number T(n,k) of set partitions of [n] having exactly k triples (t,t+1,t+2) such that t+i is in block b+i for some b; triangle T(n,k), n>=0, 0<=k<=max(0,n-2), read by rows.
|
|
3
|
|
|
|
1, 1, 2, 4, 1, 10, 4, 1, 28, 18, 5, 1, 89, 77, 30, 6, 1, 315, 345, 164, 45, 7, 1, 1233, 1617, 919, 299, 63, 8, 1, 5285, 8003, 5262, 2011, 492, 84, 9, 1, 24583, 41871, 31180, 13611, 3857, 754, 108, 10, 1, 123062, 231474, 191889, 94020, 30128, 6755, 1095, 135, 11, 1
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,3
|
|
|
LINKS
|
|
|
|
EXAMPLE
|
T(3,1) = 1: 1|2|3.
T(4,1) = 4: 12|3|4, 14|2|3, 1|24|3, 1|2|34.
T(5,1) = 18: 123|4|5, 125|3|4, 12|35|4, 12|3|45, 13|24|5, 1|23|4|5, 145|2|3, 14|25|3, 14|2|35, 14|2|3|5, 15|24|3, 1|245|3, 1|24|35, 1|24|3|5, 15|2|34, 1|25|34, 1|2|345, 1|2|34|5.
T(5,2) = 5: 12|3|4|5, 15|2|3|4, 1|25|3|4, 1|2|35|4, 1|2|3|45.
T(5,3) = 1: 1|2|3|4|5.
Triangle T(n,k) begins:
: 0 : 1;
: 1 : 1;
: 2 : 2;
: 3 : 4, 1;
: 4 : 10, 4, 1;
: 5 : 28, 18, 5, 1;
: 6 : 89, 77, 30, 6, 1;
: 7 : 315, 345, 164, 45, 7, 1;
: 8 : 1233, 1617, 919, 299, 63, 8, 1;
: 9 : 5285, 8003, 5262, 2011, 492, 84, 9, 1;
: 10 : 24583, 41871, 31180, 13611, 3857, 754, 108, 10, 1;
|
|
|
MAPLE
|
b:= proc(n, i, t, m) option remember; expand(`if`(n=0, 1, add((v->
`if`(t and v, x, 1)*b(n-1, j, v, max(m, j)))(j=i+1), j=1..m+1)))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 1, false, 0)):
seq(T(n), n=0..14);
|
|
|
MATHEMATICA
|
b[n_, i_, t_, m_] := b[n, i, t, m] = Expand[If[n==0, 1, Sum[Function[v, If[t && v, x, 1]*b[n-1, j, v, Max[m, j]]][j==i+1], {j, 1, m+1}]]]; T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, 1, False, 0]]; Table[T[n], {n, 0, 14}] // Flatten (* Jean-François Alcover, Feb 05 2017, translated from Maple *)
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn,tabf
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
