|
|
A322384
|
|
Number T(n,k) of entries in the k-th cycles of all permutations of [n] when cycles are ordered by decreasing lengths (and increasing smallest elements); triangle T(n,k), n>=1, 1<=k<=n, read by rows.
|
|
15
|
|
|
1, 3, 1, 13, 4, 1, 67, 21, 7, 1, 411, 131, 46, 11, 1, 2911, 950, 341, 101, 16, 1, 23563, 7694, 2871, 932, 197, 22, 1, 213543, 70343, 26797, 9185, 2311, 351, 29, 1, 2149927, 709015, 275353, 98317, 27568, 5119, 583, 37, 1, 23759791, 7867174, 3090544, 1141614, 343909, 73639, 10366, 916, 46, 1
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,2
|
|
LINKS
|
|
|
EXAMPLE
|
The 6 permutations of {1,2,3} are:
(1) (2) (3)
(1,2) (3)
(1,3) (2)
(2,3) (1)
(1,2,3)
(1,3,2)
so there are 13 elements in the first cycles, 4 in the second cycles and only 1 in the third cycles.
Triangle T(n,k) begins:
1;
3, 1;
13, 4, 1;
67, 21, 7, 1;
411, 131, 46, 11, 1;
2911, 950, 341, 101, 16, 1;
23563, 7694, 2871, 932, 197, 22, 1;
213543, 70343, 26797, 9185, 2311, 351, 29, 1;
...
|
|
MAPLE
|
b:= proc(n, l) option remember; `if`(n=0, add(l[-i]*
x^i, i=1..nops(l)), add(binomial(n-1, j-1)*
b(n-j, sort([l[], j]))*(j-1)!, j=1..n))
end:
T:= n-> (p-> (seq(coeff(p, x, i), i=1..n)))(b(n, [])):
seq(T(n), n=1..12);
|
|
MATHEMATICA
|
b[n_, l_] := b[n, l] = If[n == 0, Sum[l[[-i]]*x^i, {i, 1, Length[l]}], Sum[Binomial[n-1, j-1]*b[n-j, Sort[Append[l, j]]]*(j-1)!, {j, 1, n}]];
T[n_] := CoefficientList[b[n, {}], x] // Rest;
|
|
CROSSREFS
|
|
|
KEYWORD
|
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|