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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.

%I A322384 #31 Dec 07 2021 22:44:34
%S A322384 1,3,1,13,4,1,67,21,7,1,411,131,46,11,1,2911,950,341,101,16,1,23563,
%T A322384 7694,2871,932,197,22,1,213543,70343,26797,9185,2311,351,29,1,2149927,
%U A322384 709015,275353,98317,27568,5119,583,37,1,23759791,7867174,3090544,1141614,343909,73639,10366,916,46,1
%N 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.
%H A322384 Alois P. Heinz, <a href="/A322384/b322384.txt">Rows n = 1..141, flattened</a>
%H A322384 Andrew V. Sills, <a href="https://arxiv.org/abs/1912.05306">Integer Partitions Probability Distributions</a>, arXiv:1912.05306 [math.CO], 2019.
%H A322384 Wikipedia, <a href="https://en.wikipedia.org/wiki/Permutation">Permutation</a>
%e A322384 The 6 permutations of {1,2,3} are:
%e A322384   (1)     (2) (3)
%e A322384   (1,2)   (3)
%e A322384   (1,3)   (2)
%e A322384   (2,3)   (1)
%e A322384   (1,2,3)
%e A322384   (1,3,2)
%e A322384 so there are 13 elements in the first cycles, 4 in the second cycles and only 1 in the third cycles.
%e A322384 Triangle T(n,k) begins:
%e A322384        1;
%e A322384        3,     1;
%e A322384       13,     4,     1;
%e A322384       67,    21,     7,    1;
%e A322384      411,   131,    46,   11,    1;
%e A322384     2911,   950,   341,  101,   16,   1;
%e A322384    23563,  7694,  2871,  932,  197,  22,  1;
%e A322384   213543, 70343, 26797, 9185, 2311, 351, 29, 1;
%e A322384   ...
%p A322384 b:= proc(n, l) option remember; `if`(n=0, add(l[-i]*
%p A322384       x^i, i=1..nops(l)), add(binomial(n-1, j-1)*
%p A322384       b(n-j, sort([l[], j]))*(j-1)!, j=1..n))
%p A322384     end:
%p A322384 T:= n-> (p-> (seq(coeff(p, x, i), i=1..n)))(b(n, [])):
%p A322384 seq(T(n), n=1..12);
%t A322384 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 A322384 T[n_] := CoefficientList[b[n, {}], x] // Rest;
%t A322384 Array[T, 12] // Flatten  (* _Jean-François Alcover_, Feb 26 2020, after _Alois P. Heinz_ *)
%Y A322384 Columns k=1-10 give: A028418, A332851, A332852, A332853, A332854, A332855, A332856, A332857, A332858, A332859.
%Y A322384 Row sums give A001563.
%Y A322384 T(2n,n) gives A332928.
%Y A322384 Cf. A185105, A322383.
%K A322384 nonn,tabl
%O A322384 1,2
%A A322384 _Alois P. Heinz_, Dec 05 2018