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

%I A338001 #18 Nov 18 2020 12:35:53
%S A338001 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,
%T A338001 18,6,18,8,8,5,6,0,5040,48,48,240,18,24,12,72,12,8,24,10,10,6,7,0,
%U A338001 40320,384,96,192,1440,36,36,24,36,360,32,12,32,16,96,15,10,30,12,12,7,8
%N A338001 Irregular triangle read by rows, a refinement of A271708.
%C A338001 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.
%C A338001 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.
%H A338001 S. W. Golomb and P. Gaal, <a href="https://doi.org/10.1006/aama.1997.0567">On the number of permutations of n objects with greatest cycle length k</a>, Adv. in Appl. Math., 20(1), 1998, 98-107.
%e A338001 Triangle rows start:
%e A338001 0: [1];
%e A338001 1: [0], [1];
%e A338001 2: [0], [2],    [2];
%e A338001 3: [0], [6],    [2],           [3];
%e A338001 4: [0], [24],   [8, 4],        [3],              [4];
%e A338001 5: [0], [120],  [8, 12],       [6, 6],           [4],         [5];
%e A338001 6: [0], [720],  [48, 16, 48],  [18, 6, 18],      [8, 8],      [5],      [6];
%e A338001 7: [0], [5040], [48, 48, 240], [18, 24, 12, 72], [12, 8, 24], [10, 10], [6], [7];
%e A338001 .
%e A338001 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.
%o A338001 (SageMath)
%o A338001 def A338001(n):
%o A338001     R = []
%o A338001     for k in (0..n):
%o A338001         P = Partitions(n, max_part=k, inner=[k])
%o A338001         q = [p.aut() for p in P]
%o A338001         R.append(q if q != [] else [0])
%o A338001     return flatten(R)
%o A338001 for n in (0..7): print(A338001(n))
%Y A338001 Cf. A271708, A110143 (row sums), A052810 (row length), A126074, A036039.
%K A338001 nonn,tabf
%O A338001 0,5
%A A338001 _Peter Luschny_, Nov 13 2020