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NonCrossing Partitions For Classical Reflection Groups
 Discrete Math
, 1996
"... We introduce analogues of the lattice of noncrossing set partitions for the classical reflection groups of type B and D. The type B analogues (first considered by Montenegro in a different guise) turn out to be as wellbehaved as the original noncrossing set partitions, and the type D analogues ..."
Abstract

Cited by 110 (5 self)
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We introduce analogues of the lattice of noncrossing set partitions for the classical reflection groups of type B and D. The type B analogues (first considered by Montenegro in a different guise) turn out to be as wellbehaved as the original noncrossing set partitions, and the type D analogues almost as wellbehaved. In both cases, they are ELlabellable ranked lattices with symmetric chain decompositions (selfdual for type B), whose rankgenerating functions, zeta polynomials, rankselected chain numbers have simple closed forms.
Higher dimensional algebra III: ncategories and the algebra of opetopes
, 1997
"... We give a definition of weak ncategories based on the theory of operads. We work with operads having an arbitrary set S of types, or ‘Soperads’, and given such an operad O, we denote its set of operations by elt(O). Then for any Soperad O there is an elt(O)operad O + whose algebras are Soperads ..."
Abstract

Cited by 74 (6 self)
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We give a definition of weak ncategories based on the theory of operads. We work with operads having an arbitrary set S of types, or ‘Soperads’, and given such an operad O, we denote its set of operations by elt(O). Then for any Soperad O there is an elt(O)operad O + whose algebras are Soperads over O. Letting I be the initial operad with a oneelement set of types, and defining I 0+ = I, I (i+1)+ = (I i+) +, we call the operations of I (n−1)+ the ‘ndimensional opetopes’. Opetopes form a category, and presheaves on this category are called ‘opetopic sets’. A weak ncategory is defined as an opetopic set with certain properties, in a manner reminiscent of Street’s simplicial approach to weak ωcategories. In a similar manner, starting from an arbitrary operad O instead of I, we define ‘ncoherent Oalgebras’, which are n times categorified analogs of algebras of O. Examples include ‘monoidal ncategories’, ‘stable ncategories’, ‘virtual nfunctors ’ and ‘representable nprestacks’. We also describe how ncoherent Oalgebra objects may be defined in any (n + 1)coherent Oalgebra.