login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 59th year, we have over 358,000 sequences, and we’ve crossed 10,300 citations (which often say “discovered thanks to the OEIS”).

Other ways to Give
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A322383 Number T(n,k) of entries in the k-th cycles of all permutations of [n] when cycles are ordered by increasing lengths (and increasing smallest elements); triangle T(n,k), n>=1, 1<=k<=n, read by rows. 15
1, 3, 1, 10, 7, 1, 45, 37, 13, 1, 236, 241, 101, 21, 1, 1505, 1661, 896, 226, 31, 1, 10914, 13301, 7967, 2612, 442, 43, 1, 90601, 117209, 78205, 29261, 6441, 785, 57, 1, 837304, 1150297, 827521, 346453, 88909, 14065, 1297, 73, 1, 8610129, 12314329, 9507454, 4338214, 1253104, 234646, 28006, 2026, 91, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

LINKS

Alois P. Heinz, Rows n = 1..141, flattened

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)

(2) (1,3)

(3) (1,2)

(1,2,3)

(1,3,2)

so there are 10 elements in the first cycles, 7 in the second cycles and only 1 in the third cycles.

Triangle T(n,k) begins:

1;

3, 1;

10, 7, 1;

45, 37, 13, 1;

236, 241, 101, 21, 1;

1505, 1661, 896, 226, 31, 1;

10914, 13301, 7967, 2612, 442, 43, 1;

90601, 117209, 78205, 29261, 6441, 785, 57, 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, l.x^Range[Length[l]], Sum[Binomial[n - 1, j - 1] b[n - j, Sort[Append[l, j]]] (j - 1)!, {j, 1, n}]];

T[n_] := Rest @ CoefficientList[b[n, {}], x];

Array[T, 12] // Flatten (* Jean-François Alcover, Mar 03 2020, after Alois P. Heinz *)

CROSSREFS

Columns k=1-10 give: A028417, A332906, A332907, A332908, A332909, A332910, A332911, A332912, A332913, A332914.

Row sums give A001563.

Cf. A185105, A322384.

Sequence in context: A117207 A046658 A124574 * A295856 A052964 A084178

Adjacent sequences: A322380 A322381 A322382 * A322384 A322385 A322386

KEYWORD

nonn,tabl

AUTHOR

Alois P. Heinz, Dec 05 2018

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified December 6 10:03 EST 2022. Contains 358630 sequences. (Running on oeis4.)