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%I #28 Oct 06 2021 19:56:00
%S 1,3,1,10,7,1,45,37,13,1,236,241,101,21,1,1505,1661,896,226,31,1,
%T 10914,13301,7967,2612,442,43,1,90601,117209,78205,29261,6441,785,57,
%U 1,837304,1150297,827521,346453,88909,14065,1297,73,1,8610129,12314329,9507454,4338214,1253104,234646,28006,2026,91,1
%N 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.
%H Alois P. Heinz, <a href="/A322383/b322383.txt">Rows n = 1..141, flattened</a>
%H Andrew V. Sills, <a href="https://arxiv.org/abs/1912.05306">Integer Partitions Probability Distributions</a>, arXiv:1912.05306 [math.CO], 2019.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Permutation">Permutation</a>
%e The 6 permutations of {1,2,3} are:
%e (1) (2) (3)
%e (1) (2,3)
%e (2) (1,3)
%e (3) (1,2)
%e (1,2,3)
%e (1,3,2)
%e so there are 10 elements in the first cycles, 7 in the second cycles and only 1 in the third cycles.
%e Triangle T(n,k) begins:
%e 1;
%e 3, 1;
%e 10, 7, 1;
%e 45, 37, 13, 1;
%e 236, 241, 101, 21, 1;
%e 1505, 1661, 896, 226, 31, 1;
%e 10914, 13301, 7967, 2612, 442, 43, 1;
%e 90601, 117209, 78205, 29261, 6441, 785, 57, 1;
%e ...
%p b:= proc(n, l) option remember; `if`(n=0, add(l[i]*
%p x^i, i=1..nops(l)), add(binomial(n-1, j-1)*
%p b(n-j, sort([l[], j]))*(j-1)!, j=1..n))
%p end:
%p T:= n-> (p-> (seq(coeff(p, x, i), i=1..n)))(b(n, [])):
%p seq(T(n), n=1..12);
%t 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 T[n_] := Rest @ CoefficientList[b[n, {}], x];
%t Array[T, 12] // Flatten (* _Jean-François Alcover_, Mar 03 2020, after _Alois P. Heinz_ *)
%Y Columns k=1-10 give: A028417, A332906, A332907, A332908, A332909, A332910, A332911, A332912, A332913, A332914.
%Y Row sums give A001563.
%Y Cf. A185105, A322384.
%K nonn,tabl
%O 1,2
%A _Alois P. Heinz_, Dec 05 2018
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