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 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

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