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A338001 Irregular triangle read by rows, a refinement of A271708. 1
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

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.

LINKS

Table of n, a(n) for n=0..74.

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.

EXAMPLE

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.

PROG

(SageMath)

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))

CROSSREFS

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

Sequence in context: A185896 A076256 A127467 * A271708 A284983 A140333

Adjacent sequences:  A337998 A337999 A338000 * A338002 A338003 A338004

KEYWORD

nonn,tabf

AUTHOR

Peter Luschny, Nov 13 2020

STATUS

approved

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Last modified September 27 10:28 EDT 2021. Contains 347689 sequences. (Running on oeis4.)