1,3

|a(n,k)| = number of sets of permutations of {1,...,n} with k total cycles.

Comments from David Callan (callan(AT)stat.wisc.edu), Sep 20 2007: (Start) |a(n,k)| = stirling1(n, k) * bell(k) counts the above sets of permutations. To see this, recall that stirling1(n, k) is the number of permutations of [n]={1,...,n} with k cycles and bell(k) is the number of set partitions of [k].

Given such a permutation and set partition, write the permutation in standard cycle form (smallest entry first in each cycle and first entries decreasing left to right). For example, with n=15 and k=6, {{10}, {6, 11}, {5, 7, 15}, {3, 13, 12, 8}, {2, 14, 9}, {1, 4}} is in this standard cycle form.

Then combine cycles as specified by the partition to form a set of lists. For example, the partition 156-24-3 would yield {{10, 2, 14, 9, 1, 4}, {6, 11, 3, 13, 12, 8}, {5, 7, 15}}. The original first entries are now the record left-to-right lows.

Finally, apply to each list the well known transformation that sends # record lows to # cycles. The example yields {{4, 14, 1, 2, 10, 9}, {13, 11, 3, 6, 8, 12}, {7, 15, 5}}. This is a bijection to sets of lists (i.e. permutations) with a total of k cycles, as required. (End)

E.g.f.: exp((1+x)^y-1).

a(n, k) = stirling1(n, k) * bell(k). - Vladeta Jovovic (vladeta(AT)eunet.rs), Feb 01 2003

1; -1,2; 2,-6,5; -6,22,-30,15; 24,-100,175,-150,52; ...

|a(3,2)|=6 because (12)(3), (12)|(3), (13)(2), (13)|(2), (23)(1), (23)|(1).

Row sums of |a(n, k)| give A000262. Cf. A008275, A008297, A055925.

Sequence in context: A122070 A144160 A115255 this_sequence A156563 A054917 A111419

Adjacent sequences: A055921 A055922 A055923 this_sequence A055925 A055926 A055927

sign,tabl

Wouter Meeussen (wouter.meeussen(AT)pandora.be), Christian G. Bower (bowerc(AT)usa.net), Jul 06 2000