Take Q = ( Q 1 , Q 2 , ? ) to be an exponential structure and M ( n ) to be the number of minimal elements of Q n where M ( 0 ) = 1 . Then a sequence of numbers { r n ( Q n ) } n ? 1 is defined by the equation ? n ? 1 r n ( Q n ) z n n ! M ( n ) = - log ( ? n ? 0 ( - 1 ) n z n n ! M ( n ) ) . Let Q ? n denote the poset Q n with a 0 ? adjoined and let 1 ? denote the unique maximal element in the poset Q n . Furthermore, let µ Q n be the Mobius function on the poset Q ? n . Stanley proved that r n ( Q n ) = ( - 1 ) n µ Q n ( 0 ? , 1 ? ) . This implies that the numbers r n ( Q n ) are integers. In this paper, we study the cases Q n = ? n ( r ) and Q n = Q n ( r ) where ? n ( r ) and Q n ( r ) are posets, respectively, of set partitions of r n whose block sizes are divisible by r and of r -partitions of n . In both cases we prove that r n ( ? n ( r ) ) and r n ( Q n ( r ) ) enumerate the pyramids by applying the Cartier-Foata monoid identity and further prove that r n ( ? n ( r ) ) is the generalized Euler number E r n - 1 and that r n ( Q n ( 2 ) ) is the number of complete non-ambiguous trees of size 2 n - 1 by bijections. This gives a new proof of Welker's theorem that r n ( ? n ( r ) ) = E r n - 1 and implies the construction of r -dimensional complete non-ambiguous trees. As a bonus of applying the theory of heaps, we establish a bijection between the set of complete non-ambiguous forests and the set of pairs of permutations with no common rise. This answers an open question raised by Aval et?al.
['Emma Yu Jin']
Heaps and two exponential structures