A332907 -id:A332907 - cp's OEIS frontend

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.

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

Offset: 1

Alois P. Heinz, Dec 05 2018

			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;
  ...

Alois P. Heinz, Rows n = 1..141, flattened
Andrew V. Sills, Integer Partitions Probability Distributions, arXiv:1912.05306 [math.CO], 2019.
Wikipedia, Permutation

Columns k=1-10 give: A028417, A332906, A332907, A332908, A332909, A332910, A332911, A332912, A332913, A332914.
Row sums give A001563.
Cf. A185105, A322384.

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);

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 *)

A350016 Irregular triangle read by rows: T(n,k) is the number of n-permutations whose third-shortest cycle has length exactly k; n >= 0, 0 <= k <= max(0,n-2).

Original entry on oeis.org

1, 1, 2, 5, 1, 17, 1, 6, 74, 11, 15, 20, 394, 56, 60, 120, 90, 2484, 407, 525, 490, 630, 504, 18108, 3235, 4725, 2240, 4620, 4032, 3360, 149904, 29143, 40509, 27440, 26460, 33264, 30240, 25920, 1389456, 291394, 398790, 319760, 163800, 302400, 277200, 259200, 226800

Offset: 0

Steven Finch, Dec 08 2021

If the permutation has no third cycle, then its third-longest cycle is defined to have length 0.

			Triangle begins:
[0]      1;
[1]      1;
[2]      2;
[3]      5,     1;
[4]     17,     1,     6;
[5]     74,    11,    15,    20;
[6]    394,    56,    60,   120,    90;
[7]   2484,   407,   525,   490,   630,   504;
[8]  18108,  3235,  4725,  2240,  4620,  4032,  3360;
[9] 149904, 29143, 40509, 27440, 26460, 33264, 30240, 25920;
...

Alois P. Heinz, Rows n = 0..142, flattened
Steven Finch, Second best, Third worst, Fourth in line, arxiv:2202.07621 [math.CO], 2022.

Column 0 gives 1 together with A000774.
Column 1 gives the column 3 of A208956.
Row sums give A000142.
Cf. A126074, A145877, A332907, A349979, A349980, A350015, A350273, A350274.

m:= infinity:
b:= proc(n, l) option remember; `if`(n=0, x^`if`(l[3]=m,
      0, l[3]), add(b(n-j, sort([l[], j])[1..3])
               *binomial(n-1, j-1)*(j-1)!, j=1..n))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, [m$3])):
seq(T(n), n=0..10);  # Alois P. Heinz, Dec 11 2021

m = Infinity;
b[n_, l_] := b[n, l] = If[n == 0, x^If[l[[3]] == m, 0, l[[3]]], Sum[b[n-j, Sort[Append[l, j]][[1;;3]]]*Binomial[n - 1, j - 1]*(j - 1)!, {j, 1, n}]];
T[n_] := With[{p = b[n, {m, m, m}]}, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]];
Table[T[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Dec 28 2021, after Alois P. Heinz *)

Sum_{k=0..n-2} k * T(n,k) = A332907(n) for n >= 3. - Alois P. Heinz, Dec 12 2021

cp's OEIS Frontend

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.

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