37,249 research outputs found
Counting Labelled Trees with Given Indegree Sequence
For a labelled tree on the vertex set , define the
direction of each edge to be if . The indegree sequence of
can be considered as a partition . The enumeration of
trees with a given indegree sequence arises in counting secant planes of curves
in projective spaces. Recently Ethan Cotterill conjectured a formula for the
number of trees on with indegree sequence corresponding to a partition
. In this paper we give two proofs of Cotterill's conjecture: one is
`semi-combinatorial" based on induction, the other is a bijective proof.Comment: 10 page
Counting Humps in Motzkin paths
In this paper we study the number of humps (peaks) in Dyck, Motzkin and
Schr\"{o}der paths. Recently A. Regev noticed that the number of peaks in all
Dyck paths of order is one half of the number of super Dyck paths of order
. He also computed the number of humps in Motzkin paths and found a similar
relation, and asked for bijective proofs. We give a bijection and prove these
results. Using this bijection we also give a new proof that the number of Dyck
paths of order with peaks is the Narayana number. By double counting
super Schr\"{o}der paths, we also get an identity involving products of
binomial coefficients.Comment: 8 pages, 2 Figure
Synthesis, Characterization, and Properties of Mononuclear and Dinuclear Ruthenium(II) Complexes Containing Phenanthroline and Chlorophenanthroline
The study of photophysical and photochemical properties of ruthenium complexes is of great interest for fundamental practical reasons. Ruthenium complexes have been investigated for use in artificial photosynthesis. This paper deals with the synthesis and spectroscopic investigation of custom-designed ruthenium complexes containing phenanthroline and chloro-phenanthroline ligands. These complexes maybe useful for biological electron-transfer studies. The heteroleptic ruthenium monomer complex Ru(phen)2(Cl-phen) (where phen = 1,10-phenanthroline and Cl-phen=5-chloro-1,10-phenanthroline) was prepared in a two-step procedure previously developed in our laboratory. This monomer complex was used to prepare the ruthenium homometallic dimer complex, (phen)2Ru(phen-phen)Ru(phen)2, by utilizing the Ni-catalyzed coupling reaction. Both complexes were purified by extensive column chromatography. The identity and the integrity of the monomer complex were confirmed by elemental analysis. The calculated and the experimental values for the elemental analysis were in good agreement for the monomer complex. UV/Vis absorption spectroscopy, emission spectroscopy, and cyclic voltammetry were used to investigate the properties of both the complexes
Acoustic detection of air shower cores
At an altitude of 1890m, a pre-test with an Air shower (AS) core selector and a small acoustic array set up in an anechoic pool with a volume of 20x7x7 cu m was performed, beginning in Aug. 1984. In analyzing the waveforms recorded during the effective working time of 186 hrs, three acoustic signals which cannot be explained as from any source other than AS cores were obtained, and an estimation of related parameters was made
- …