login
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 largest part 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.
3

%I #13 Mar 08 2020 10:13:36

%S 1,0,1,0,2,2,0,6,2,3,0,24,12,3,4,0,120,20,12,4,5,0,720,112,42,16,5,6,

%T 0,5040,336,126,44,20,6,7,0,40320,2112,492,188,55,24,7,8,0,362880,

%U 11712,2802,640,215,66,28,8,9,0,3628800,92160,16938,3624,830,258,77,32,9,10

%N 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 largest part 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.

%C Also T(n,k) = Sum_{p in P(n,k)} Cen(p) where Cen(p) is the size of the centralizer of any permutation of cycle type p.

%e Triangle starts:

%e [1]

%e [0, 1]

%e [0, 2, 2]

%e [0, 6, 2, 3]

%e [0, 24, 12, 3, 4]

%e [0, 120, 20, 12, 4, 5]

%e [0, 720, 112, 42, 16, 5, 6]

%e [0, 5040, 336, 126, 44, 20, 6, 7]

%e [0, 40320, 2112, 492, 188, 55, 24, 7, 8]

%o (Sage)

%o def A271708(n,k):

%o P = Partitions(n, max_part=k, inner=[k])

%o return sum([p.aut() for p in P])

%o for n in (0..9): print([A271708(n,k) for k in (0..n)])

%Y Cf. A110143 (row sums), A126074.

%K nonn,tabl

%O 0,5

%A _Peter Luschny_, Apr 17 2016