We give a direct combinatorial proof of the known fact that the 3-cube has 384 spanning trees, using an “edge slide” operation on spanning trees. This gives an answer in the case n = 3 to a question implicitly raised by Stanley. Our argument also gives a bijective proof of the n = 3 case of a weighted count of the spanning trees of the n-cube due to Martin and Reiner.

Whereas Stanley, Proctor and others studied Sperner properties of the poset L(k; m) we consider shadows of left-compressed sets and derive asymptotic bounds on their sizes. ? 1999 Elsevier Science B.V. All rights reserved.

2010 has been an interesting year for abortion provision. Early abortion, in the first trimester of pregnancy, is becoming increasingly acceptable and accessible, as reflected by liberalising laws and practice throughout the world. In Spain, much discussion has followed the implementation of a new law allowing abortion without restriction in the first 14 weeks. As Stanley Henshaw of the US Gutt...

G. E. Andrews has established a refinement of the generating function for partitions π according to the numbers O(π) and O(π′) of odd parts in π and the conjugate of π, respectively. In this paper, we derive a refined generating function for partitions into at most M parts less than or equal to N , which is a finite case of Andrew’s refinement.

sance writings, from scholars like Hostiensis (ca.1200–71), Oresme (1320–82) and Copernicus (1473–1543), and from mint officials like Henri Poullain (1612). As we have just seen, Aristophanes, in the 5th century B.C.E., had already anticipated the mechanism. And it was the British economist, Stanley Jevons, who originated the expression “bad money drives out good money” in his 1875 book, Money ...

The lattice of noncrossing partitions can be embedded into the Cayley graph of the symmetric group. This allows us to rederive connections between noncrossing partitions and parking functions. We use an analogous embedding for type B noncrossing partitions in order to answer a question raised by R. Stanley on the edge labeling of the type B non-crossing partitions lattice.

Let Gn denote the set of lattice paths from (0, 0) to (n, n) with steps of the form (i, j) where i and j are nonnegative integers, not both zero. Let Dn denote the set of paths in Gn with steps restricted to (1, 0), (0, 1), (1, 1), the so-called Delannoy paths. Stanley has shown that |Gn| = 2n−1|Dn| and Sulanke has given a bijective proof. Here we give a simple statistic on Gn that is uniformly...

Louis Solomon showed that the group algebra of the symmetric group Sn has a subalgebra called the descent algebra, generated by sums of permutations with a given descent set. In fact, he showed that every Coxeter group has something which can be called a descent algebra. For any Coxeter group that is also a Weyl group, Paola Cellini proved the existence of a different, commutative subalgebra of...