1,2

The strategic behavior of players depends crucially on the coalition structures of a game.

The sequence is not a duplicate of A005493. In particular A005493 (Kitaev in Sem. Loth., see Reference) presents the number of n-permutations that avoids the generalized pattern whose k rightmost letters form an increasing subword. The sequence here measures the number of embedded coalitions in a n-person game. Moreover, one can show that there is a relationship between A138378 and A005493 as follows: Y(n) from A138378 collapses into B(n-1) from A005493 which is P(n+1,2) with k=2 in Kitaev's article. [D. Yeung, priv. commun. May 20 2008] - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), May 20 2008

DAVID W. K. YEUNG, 2008, Recursive sequence identifying the number of embedded coalitions, International Game Theory Review 10(1), 129-136.

E. T. Bell, 1934, Exponential numbers, American Mathematical Monthly 41,411-419.

Conway, J. H. and Guy, R. K. [1995] The Book of Numbers (Springer-Verlag, NewYork).

DAVID W. K. YEUNG, Perfect Numbers, International Game Theory Review 10(2008), 129-136.

a(1) = combination(1,0) = 1, a(2) = combination(2,1) + combination(2,0)= 3, a(n) = {SUM(i=2 to n-1) combination(n,i)} * {SUM(j=1 to i-1) a(n)} + SUM(i=0 to n-1) combination(n,i), for n > 2.

a(1) = combination(1,0) = 1,

a(2) = combination(2,1) + combination(2,0)= 3,

a(3) = combination(3,2)* a(1) + combination(3,2) + combination(3,1) + combination(3,0)= 10,

a(4) = combination(4,3)* {a(1) + a(2)} + combination(4,2)* a(1) + combination7(4,3)combination(4,2) + combination(4,1) + combination(4,0)= 37,

a(5) = combination(5,4)* {a(1) + a(2) + a(3)} combination(5,3)* {a(1) + a(2)} + combination(5,2)* a(1) + combination(5,4) + combination(5,3) + combination(5,2) + combination(5,1) + combination(5,0)= 151.

Cf. A138379.

Sequence in context: A044048 A086444 A064613 this_sequence A005493 A123636 A092816

Adjacent sequences: A138375 A138376 A138377 this_sequence A138379 A138380 A138381

easy,nonn

David Yeung (wkyeung(AT)hkbu.edu.hk), May 08 2008