A271708 -id:A271708 - cp's OEIS frontend

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

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))

A271707 Triangle read by rows, T(n,k) = Sum_{p in P(n,k)} Aut(p) where P(n,k) are the partitions of n with length k and Aut(p) = 1^j[1]j[1]!...n^j[n]j[n]! where j[m] is the number of parts in the partition p equal to m; for n>=0 and 0<=k<=n.

Original entry on oeis.org

1, 0, 1, 0, 2, 2, 0, 3, 2, 6, 0, 4, 11, 4, 24, 0, 5, 10, 14, 12, 120, 0, 6, 31, 62, 34, 48, 720, 0, 7, 28, 60, 84, 120, 240, 5040, 0, 8, 66, 102, 490, 228, 552, 1440, 40320, 0, 9, 60, 299, 292, 708, 912, 3120, 10080, 362880, 0, 10, 120, 282, 722, 4396, 2136, 4752, 20880, 80640, 3628800

Offset: 0

Peter Luschny, Apr 17 2016

S(n,k) = Sum_{p in P(n,k)} n!/Aut(p) are the Stirling cycle numbers A132393.

			Triangle starts:
[1]
[0, 1]
[0, 2, 2]
[0, 3, 2, 6]
[0, 4, 11, 4, 24]
[0, 5, 10, 14, 12, 120]
[0, 6, 31, 62, 34, 48, 720]
[0, 7, 28, 60, 84, 120, 240, 5040]

Cf. A110143 (row sums), A132393, A271708.

def A271707(n,k):
    P = Partitions(n, length=k)
    return sum(p.aut() for p in P)
for n in (0..10): print([A271707(n,k) for k in (0..n)])

cp's OEIS Frontend

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

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A271707 Triangle read by rows, T(n,k) = Sum_{p in P(n,k)} Aut(p) where P(n,k) are the partitions of n with length k and Aut(p) = 1^j[1]j[1]!...n^j[n]j[n]! where j[m] is the number of parts in the partition p equal to m; for n>=0 and 0<=k<=n.

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Sage

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

SageMath

A271707 Triangle read by rows, T(n,k) = Sum_{p in P(n,k)} Aut(p) where P(n,k) are the partitions of n with length k and Aut(p) = 1^j[1]*j[1]!*...*n^j[n]*j[n]! where j[m] is the number of parts in the partition p equal to m; for n>=0 and 0<=k<=n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Sage

A271707 Triangle read by rows, T(n,k) = Sum_{p in P(n,k)} Aut(p) where P(n,k) are the partitions of n with length k and Aut(p) = 1^j[1]j[1]!...n^j[n]j[n]! where j[m] is the number of parts in the partition p equal to m; for n>=0 and 0<=k<=n.