The r-cubical Lattice and a Generalization of the cd-index

Abstract

In this paper we generalize the cd -index of the cubical lattice to an r -cd -index , which we denote by Õ ( r ) . The coef ficients of Õ ( r ) enumerate augmented Andre ´ r -signed permutations , a generalization of Purtill's work relating the cd -index of the cubical lattice and signed Andre ´ permutations . As an application we use the r -cd -index to determine that the extremal configuration which maximizes the Mo ¨ bius function of arbitrary rank selections , where all the r i 's are greater than one , is the odd alternating ranks , ͕ 1 , 3 , 5 , . . . ͖ .÷ 1996 Academic Press Limited . I NTRODUCTIONThe main purpose of this paper is to develop a generalization of the cd -index for the r -cubical lattice C r . This lattice is a natural generalization of the cubical lattice , that is , the face lattice of a cube . The cubical lattice of order n may be described by taking the n th power of the poset M 2 in Figure 1 and then adjoining a minimal element . The r -cubical lattice is constructed in a similar manner , where we instead take a product of posets M r of the form represented in Figure 1 . Such a lattice was first studied by Metropolis , Rota , Strehl and White in [12] . They were interested in Dilworth decompositions of the r -cubical lattice .The ab -index is a non-commutative polynomial which encodes the Mo ¨ bius function of rank selections from a poset P , i . e ., its flag h -vector or beta invariant . Equivalently , the ab -index encodes the flag f -vector of P . Fine observed that when P is an Eulerian poset , the ab -index can be written uniquely in the variables c ϭ a ϩ b and d ϭ ab ϩ ba (see [2]) . This new polynomial is called the cd -index .The importance of the cd -index is that it explicitly describes the generalized Dehn -Sommerville equations , also known as the Bayer -Billera relations [1] . Purtill obtained recursion formulas for the cd -index of the boolean algebra B n and the cubical lattice C n . In order to do , he showed that the coef ficients of each cd -index enumerate Andre ´ permutations and signed Andre ´ permutations , respectively . Andre ´ permutations were first studied by Foata and Schu ¨ tzenberger [8 , 9] . We show that these two recurrences are easy to prove by using the standard R -labeling of B n and C n (see the arguments before equations (1) and (3)) .As a corollary , Purtill concluded that the cd -index of B n and C n have positive coef ficients . Using a shelling argument , Stanley extended this result to showing that the cd -index has non-negative coef ficients when P is the face poset of a shellable regular CW -sphere . This class of posets includes face lattices of convex polytopes .Although the r -cubical lattice is not an Eulerian poset , we are still able to form its ab -index . We give a recursion for its ab -index in terms of the non-commutative variables c s ϭ a ϩ ( s Ϫ 1) ؒ b , d s ϭ ab ϩ ( s Ϫ 1) ؒ ba , c and d . Since this recursion coincides with Purtill's cd -index recurrence for the cubical lattice , that is , when r ϭ (2 , . . . , 2) , we call it ...

On Flag Vectors, the Dowling Lattice, and Braid Arrangements

“…For a maximal chain c = {0 From this result we obtain the following corollary, which was observed in [15]. An EL-labeling of a graded poset P is an R-labeling such that in each interval [x, y] the unique rising chain is lexicographically least among all chains in the interval [x, y].…”

Untitled

“…Examples of such posets and permutation sets are the Boolean algebra and André permutations, and the cubical lattice and signed André permutations. See [8,17]. For more refined identities using such a poset-permutation set correspondence, see [6,11,20].…”

“…We will assume that the reader is familiar with R-labelings. Otherwise, see [8] or [19,Section 3.13]. …”

“…Similarly, it is also a refined enumeration of simsun permutations, first defined by Simion and Sundaram [22,23]. Another known example of a poset-permutations pair is the cubical lattice and signed André permutations [8,17]. This motivates us to ask the following question.…”

The generating function for the Dirichlet series $L_{m}(s)$

“….) has been given implicitly by Ehrenborg and Readdy [5] in terms of nonaugmented André R-signed permutations on [n] with R = (4, 4, . .…”