{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,11]],"date-time":"2026-03-11T17:38:03Z","timestamp":1773250683863,"version":"3.50.1"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>Let $[u,v]$ be a Bruhat interval of a Coxeter group such that the Bruhat graph $BG(u,v)$ of $[u,v]$ is isomorphic to a Boolean lattice. In this paper, we provide a combinatorial explanation for the coefficients of the complete cd-index of $[u,v]$. Since in this case the complete cd-index and the cd-index of $[u,v]$ coincide, we also obtain a new combinatorial interpretation for the coefficients of the cd-index of Boolean lattices. To this end, we label an edge in $BG(u,v)$ by a pair of nonnegative integers and show that there is a one-to-one correspondence between such sequences of nonnegative integer pairs and Bruhat paths in $BG(u,v)$. Based on this labeling, we construct a flip $\\mathcal{F}$ on the set of Bruhat paths in $BG(u,v)$, which is an involution that changes the ascent-descent sequence of a path. Then we show that the flip $\\mathcal{F}$ is compatible with any given reflection order and also satisfies the flip condition for any cd-monomial $M$. Thus by results of Karu, the coefficient of $M$ enumerates certain Bruhat paths in $BG(u,v)$, and so can be interpreted as the number of certain sequences of nonnegative integer pairs. Moreover, we give two applications of the flip $\\mathcal{F}$. We enumerate the number of cd-monomials in the complete cd-index of $[u,v]$ in terms of Entringer numbers, which are refined enumerations of Euler numbers. We also give a refined enumeration of the coefficient of d${}^n$ in terms of Poupard numbers, and so obtain new combinatorial interpretations for Poupard numbers and reduced tangent numbers.<\/jats:p>","DOI":"10.37236\/5100","type":"journal-article","created":{"date-parts":[[2020,1,10]],"date-time":"2020-01-10T23:46:51Z","timestamp":1578700011000},"source":"Crossref","is-referenced-by-count":3,"title":["The Complete cd-Index of Boolean Lattices"],"prefix":"10.37236","volume":"22","author":[{"given":"Neil J.Y.","family":"Fan","sequence":"first","affiliation":[]},{"given":"Liao","family":"He","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2015,6,3]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v22i2p45\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v22i2p45\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T10:15:55Z","timestamp":1579256155000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v22i2p45"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2015,6,3]]},"references-count":0,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2015,4,14]]}},"URL":"https:\/\/doi.org\/10.37236\/5100","relation":{},"ISSN":["1077-8926"],"issn-type":[{"value":"1077-8926","type":"electronic"}],"subject":[],"published":{"date-parts":[[2015,6,3]]},"article-number":"P2.45"}}