On the (co)homology of the poset of weighted partitions

Abstract

We consider the poset of weighted partitions Π w n , introduced by Dotsenko and Khoroshkin in their study of a certain pair of dual operads. The maximal intervals of Π w n provide a generalization of the lattice Π n of partitions, which we show possesses many of the well-known properties of Π n . In particular, we prove these intervals are EL-shellable, we show that the Möbius invariant of each maximal interval is given up to sign by the number of rooted trees on node set {1, 2, . . . , n} having a fixed number of descents, we find combinatorial bases for homology and cohomology, and we give an explicit sign twisted S n -module isomorphism from cohomology to the multilinear component of the free Lie algebra with two compatible brackets. We also show that the characteristic polynomial of Π w n has a nice factorization analogous to that of Π n .

On the free Lie algebra with multiple brackets

“…The In [22] the authors found a nice EL-labeling of Π w n ∪ {1} that generalized a classical EL-labeling of Π n due to Björner and Stanley (see [5]). An EL-labeling of a poset (defined in Section 3.2) is a labeling of the edges of the Hasse diagram of the poset that satisfies certain requirements.…”

“…The symmetric group acts naturally on each Lie 2 (n, i) and on each open interval (0, [n] i ). In [22] González D'León and Wachs give an explicit S n -module isomorphism which establishes 3) and the isomorphism reduces to the one in [41]. In [22] bases for H n−3 ((0, [n] i )) and for Lie 2 (n, i) are constructed generalizing the classical Lyndon tree basis and the comb basis for H n−3 (Π n ) and Lie(n).…”

“…In [22] González D'León and Wachs give an explicit S n -module isomorphism which establishes 3) and the isomorphism reduces to the one in [41]. In [22] bases for H n−3 ((0, [n] i )) and for Lie 2 (n, i) are constructed generalizing the classical Lyndon tree basis and the comb basis for H n−3 (Π n ) and Lie(n). In particular, the general Lyndon basis is obtained from the ascent-free maximal chains of the EL-labeling of Theorem 1.1.…”

“…(See Theorem 3.2, Corollary 3.5 and Theorem 3.6 [22].) The poset Π w n := Π w n ∪ {1} is EL-shellable and hence Cohen-Macaulay.…”

“…In particular, the general Lyndon basis is obtained from the ascent-free maximal chains of the EL-labeling of Theorem 1.1. The authors of [22] also define a basis for H n−3 ((0, [n] i )) in terms of labeled rooted trees that generalizes the Björner NBC basis for homology of Π n (see [6,Proposition 2.2]). …”