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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
Andrew V. Sills, Integer Partitions Probability Distributions, arXiv:1912.05306 [math.CO], 2019.
Wikipedia, Permutation
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;
Array[T, 12] // Flatten (* Jean-François Alcover, Feb 26 2020, after Alois P. Heinz *)
CROSSREFS
Row sums give A001563.
T(2n,n) gives A332928.
Sequence in context: A295827 A277197 A297898 * A360088 A113139 A266577
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Dec 05 2018
STATUS
approved

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Last modified May 21 05:34 EDT 2024. Contains 372728 sequences. (Running on oeis4.)