1,4

Number of successive equalities s_i = s_{i+1} in a set partition {s_1, ..., s_n} of {1, ..., n}, where s_i is the subset containing i, s(1) = 1, and s(i) <= 1 + max of previous s(j)'s.

T(n,c)=number of set partitions of the set {1,2,...,n} in which the size of the block containing the element 1 is k+1. Example: T(4,2)=3 because we have 123|4, 124|3, and 1342. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Nov 10 2006

W. C. Yang, Conjectures on some sequences involving set partitions and Bell numbers, preprint, 2000.

T(n, c) = B(n - 1 - c)*binomial(n - 1, c), where T(n, c) is the number of set partitions of {1, ..., n} that have c successive equalities and B() is a Bell number.

E.g.f.: exp(exp(x)+x*y-1). - Vladeta Jovovic (vladeta(AT)Eunet.yu), Feb 13 2003

For example {1, 2, 1, 2, 2, 3} is a set partition of {1, 2, 3, 4, 5, 6} and has 1 successive equality, at i = 4.

1; 1,1; 2,2,1; 5,6,3,1; 15,20,12,4,1; 52,75,50,20,5,1; ...

with(combinat): T:=(n, c)->binomial(n-1, c)*bell(n-1-c): for n from 1 to 11 do seq(T(n, c), c=0..n-1) od; # yields sequence in triangular form - Emeric Deutsch (deutsch(AT)duke.poly.edu), Nov 10 2006

Cf. Bell numbers A000110.

Cf. A056858-A056863. Essentially same as A056860, where the rows are read from right to left.

Sequence in context: A019710 A118806 A124644 this_sequence A129100 A127082 A065052

Adjacent sequences: A056854 A056855 A056856 this_sequence A056858 A056859 A056860

easy,nonn,tabl,nice

Winston C. Yang (winston(AT)cs.wisc.edu), Aug 31 2000

More terms from David Wasserman (wasserma(AT)spawar.navy.mil), Apr 22 2002