Counting Split Interval Orders

A poset (X,) is a split interval order (a.k.a. unit bitolerance order, proper bitolerance order) if a real interval and a distinguished point in that interval can be assigned to each xX so that xy precisely when x's distinguished point precedes y's interval, and x's interval precedes y's distinguished point. For each |X|9, we count the split interval orders and identify all posets that are minimal forbidden posets for split interval orders. The paper is a companion to Counting Split Semiorders by Fishburn and Reeds (this issue).     

The d-Step Conjecture and Gaussian Elimination

The d-step conjecture is one of the fundamental open problems concerning the structure of convex polytopes. Let Δ (d,n) denote the maximum diameter of a graph of a d-polytope that has n facets. The d-step conjecture Δ (d,2d) = d is proved equivalent to the following statement: For each ``general position' real matrix M there are two matrices drawn from a finite group matrices isomorphic to the symmetric group on d letters, such that has the Gaussian elimination factorization L ^-1 U in which L and U are lower triangular and upper triangular matrices, respectively, that have positive nontriangular elements. If #(M) is the number of pairs giving a positive L ^-1 U factorization, then #(M) equals the number of d-step paths between two vertices of an associated Dantzig figure. One consequence is that #(M)≤ d!. Numerical experiments all satisfied #(M) ≥ 2 ^d-1 , including examples attaining equality for 3 ≤ d ≤ 15. The inequality #(M) ≥ 2 ^d-1 is proved for d=3. For d≥ 4, examples with #(M) =2 ^d-1 exhibit a large variety of combinatorial types of associated Dantzig figures. These experiments and other evidence suggest that the d-step conjecture may be true in all dimensions, in the strong form #(M) ≥ 2 ^d-1 . Received April 10, 1995, and in revised form August 23, 1995.     

Formation and propagation of well-defined Pd nanoparticles (PdNPs) during C–H bond functionalization of heteroarenes: are nanoparticles a moribund form of Pd or an active catalytic species?

Examination of a series of C–H bond functionalization reactions of heteroarenes (e.g., indole, benzoxazole, benzthiazole, benzimidazole and purine derivatives) mediated by Pd(OAc)₂, a commonly used C–H bond functionalization catalyst, reveals that well-defined Pd nanoparticles (PdNPs) are rapidly formed under working catalyst conditions. The PdNPs can be characterized ex situ after entrapment in a polymer matrix (polyvinylpyrrolidinone, PVP). Independently synthesized Pd(PVP)NPs are catalytically competent species, exhibiting catalyst activity commensurate with Pd(OAc)₂ in several C–H bond functionalization reactions. Across a range of reactions, Pd concentration is a common variable, which can be linked to the propagation of PdNPs under working catalytic conditions using polar solvents like DMF, DMSO and acetic acid.     

Carbon monoxide-induced N-N bond cleavage of nitrous oxide that is competitive with oxygen atom transfer to carbon monoxide as mediated by a Mo(II)/Mo(IV) catalytic cycle

In the presence of CO, facile N-N bond cleavage of N(2)O occurs at the formal Mo(II) center within coordinatively unsaturated mononuclear species derived from Cp*Mo[N((i)Pr)C(Me)N((i)Pr)](CO)(2) (Cp* = η(5)-C(5)Me(5)) (1) and {Cp*Mo[N((i)Pr)C(Me)N((i)Pr)]}(2)(μ-η(1):η(1)-N(2)) (9) under photolytic and dark conditions, respectively, to produce the nitrosyl, isocyanate complex Cp*Mo[N((i)Pr)C(Me)N((i)Pr)](κ-N-NO)(κ-N-NCO) (7). Competitive N-O bond cleavage of N(2)O proceeds under the same conditions to yield the Mo(IV) terminal metal oxo complex Cp*Mo[N((i)Pr)C(Me)N((i)Pr)](O) (3), which can be recycled to produce more 7 through oxygen-atom-transfer oxidation of CO to produce CO(2).     

Counting Split Semiorders

A poset P=(X,) is a split semiorder if a unit interval and a distinguished point in that interval can be assigned to each xX so that xy precisely when x's distinguished point precedes y's interval, and y's distinguished point follows x's interval. For each |X|10, we count the split semiorders and identify all posets that are minimal forbidden posets for split semiorders.