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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. 5
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

Alois P. Heinz, Rows n = 1..50, flattened

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);

CROSSREFS

Column k=1 gives A028418.

Row sums give A001563.

Cf. A185105, A322383.

Sequence in context: A295827 A277197 A297898 * A113139 A266577 A143411

Adjacent sequences:  A322381 A322382 A322383 * A322385 A322386 A322387

KEYWORD

nonn,tabl

AUTHOR

Alois P. Heinz, Dec 05 2018

STATUS

approved

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Last modified October 17 06:08 EDT 2019. Contains 328106 sequences. (Running on oeis4.)