Counting Labelled Trees with Given Indegree Sequence

For a labelled tree on the vertex set $[n]:=\{1,2,..., n\}$, define the direction of each edge $ij$ to be $i\to j$ if $i<j$. The indegree sequence of $T$ can be considered as a partition $\lambda \vdash n-1$. 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 $[n]$ with indegree sequence corresponding to a partition $\lambda$. 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 $n$ is one half of the number of super Dyck paths of order $n$. 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 $n$ with $k$ 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