Exponential self-convolution of numbers of preferential arrangements.
Number of compatible bipartitional relations on a set of cardinality n. -
Ralf Stephan, Apr 27 2003
With an extra 1 at the beginning, coefficients of the formal (divergent) series expansion at infinity of Sum_{k>=0} 1/binomial(x,k) = 1+1/x+2/x^2+8/x^3+... Also Sum_{k>=0} k!/x^k Product_{i=1..k-1} 1/(1-i/x) yields a generating function in 1/x. -
Roland Bacher, Nov 21 2000
a(n) is the total number of open sets summed over all chain topologies that can be placed on an n-set. A chain topology is a topology whose open sets can be totally ordered by inclusion. -
Geoffrey Critzer, Apr 06 2017
Also the number of length n + 1 sequences covering an initial interval of positive integers with no adjacent equal parts (anti-runs). For example, the a(0) = 1 through a(2) = 8 anti-runs are:
(1) (1,2) (1,2,1)
(2,1) (1,2,3)
(1,3,2)
(2,1,2)
(2,1,3)
(2,3,1)
(3,1,2)
(3,2,1)
Also the number of ordered set partitions of {1,...,n + 1} with no two successive vertices in the same block. For example, the a(0) = 1 through a(2) = 8 ordered set partitions are:
{{1}} {{1},{2}} {{1,3},{2}}
{{2},{1}} {{2},{1,3}}
{{1},{2},{3}}
{{1},{3},{2}}
{{2},{1},{3}}
{{2},{3},{1}}
{{3},{1},{2}}
{{3},{2},{1}}
(End)
a(n+1) is the n-th row sum in
A380977.
Number of surjections f with domain [n+1] and f(n+1)!=f(j) for j<n+1.
Number of (n+1)-tuples containing all elements of a set, with a unique last element.
Consider an urn with balls of pairwise different colors. a(n) is the number of (n+1)-sequences of draws with replacement completing the covering of all colors with the last draw, the number of colors running from 1 to n+1.
(End)
