A126074 -id:A126074 - cp's OEIS frontend

A330858 Triangle read by rows: T(n,k) is the number of permutations in S_n for which all cycles have length <= k.

1, 1, 2, 1, 4, 6, 1, 10, 18, 24, 1, 26, 66, 96, 120, 1, 76, 276, 456, 600, 720, 1, 232, 1212, 2472, 3480, 4320, 5040, 1, 764, 5916, 14736, 22800, 29520, 35280, 40320, 1, 2620, 31068, 92304, 164880, 225360, 277200, 322560, 362880, 1, 9496, 171576, 632736

Offset: 1

Peter Kagey, Apr 28 2020

			For n = 3 and k = 2, the T(3,2) = 4 permutations in S_3 where all cycle lengths are less than or equal to 2 are:
(1)(2)(3), (12)(3), (13)(2), and (1)(23).
Table begins:
n\k| 1    2     3     4      5      6      7      8      9
---+------------------------------------------------------
  1| 1
  2| 1    2
  3| 1    4     6
  4| 1   10    18    24
  5| 1   26    66    96    120
  6| 1   76   276   456    600    720
  7| 1  232  1212  2472   3480   4320   5040
  8| 1  764  5916 14736  22800  29520  35280  40320
  9| 1 2620 31068 92304 164880 225360 277200 322560 362880

Peter Kagey, Table of n, a(n) for n = 1..10011 (first 141 rows, flattened)
Steven Finch, Second best, Third worst, Fourth in line, arxiv:2202.07621 [math.CO], 2022.

Cf. A068424, A333726.
Columns 2-6: A000085, A057693, A070945, A070946, A070947.
T(n,floor(n/2)) gives A024168.
Cf. A126074.

T[n_, k_] := T[n, k] = If[n <= k, n!, n*T[n-1, k] - FactorialPower[n-1, k]* T[n-k-1, k]];
Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* Jean-François Alcover, Apr 28 2020 *)

T4(n, k)=if(k<1 || k>n, 0, n!/(n-k)!); \\ A068424
T(n,k) = if (n<=k, n!, n*T(n-1,k) - T4(n-1,k)*T(n-k-1,k));
tabl(nn) = for (n=1, nn, for (k=1, n, print1(T(n,k), ", ")); print); \\ Michel Marcus, May 09 2020

T(n,k) = n! if n <= k, otherwise T(n,k) = n*T(n-1,k) - A068424(n-1,k)*T(n-k-1,k).

T(n,k) = Sum_{j=1..k} A126074(n,j). - Alois P. Heinz, Jul 08 2022

A338001 Irregular triangle read by rows, a refinement of A271708.

Original entry on oeis.org

1, 0, 1, 0, 2, 2, 0, 6, 2, 3, 0, 24, 8, 4, 3, 4, 0, 120, 8, 12, 6, 6, 4, 5, 0, 720, 48, 16, 48, 18, 6, 18, 8, 8, 5, 6, 0, 5040, 48, 48, 240, 18, 24, 12, 72, 12, 8, 24, 10, 10, 6, 7, 0, 40320, 384, 96, 192, 1440, 36, 36, 24, 36, 360, 32, 12, 32, 16, 96, 15, 10, 30, 12, 12, 7, 8

Offset: 0

Peter Luschny, Nov 13 2020

Row n of the triangle gives the sizes of the centralizers of any permutation of cycle type given by the partitions of n with max. part k.

T(n, k) divides n! if k > 0 and in this case the n!/T(n, k) give, up to order, the rows of A036039.

			Triangle rows start:
0: [1];
1: [0], [1];
2: [0], [2],    [2];
3: [0], [6],    [2],           [3];
4: [0], [24],   [8, 4],        [3],              [4];
5: [0], [120],  [8, 12],       [6, 6],           [4],         [5];
6: [0], [720],  [48, 16, 48],  [18, 6, 18],      [8, 8],      [5],      [6];
7: [0], [5040], [48, 48, 240], [18, 24, 12, 72], [12, 8, 24], [10, 10], [6], [7];
.
For n = 4 the partition of 4 with cycle type [2, 2] has centralizer size 8, and the partition [2, 1, 1] has centralizer size 4. Therefore in column 2 in the above triangle the pair [8, 4] appears.

S. W. Golomb and P. Gaal, On the number of permutations of n objects with greatest cycle length k, Adv. in Appl. Math., 20(1), 1998, 98-107.

Cf. A271708, A110143 (row sums), A052810 (row length), A126074, A036039.

def A338001(n):
    R = []
    for k in (0..n):
        P = Partitions(n, max_part=k, inner=[k])
        q = [p.aut() for p in P]
        R.append(q if q != [] else [0])
    return flatten(R)
for n in (0..7): print(A338001(n))

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A330858 Triangle read by rows: T(n,k) is the number of permutations in S_n for which all cycles have length <= k.

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