Graphs, partitions and Fibonacci numbers

The Fibonacci number of a graph is the number of independent vertex subsets. In this paper, we investigate trees with large Fibonacci number. In particular, we show that all trees with n edges and Fibonacci number >2^n-1+5 have diameter ?4 and determine the order of these trees with respect to their Fibonacci numbers. Furthermore, it is shown that the average Fibonacci number of a star-like tree (i.e. diameter ?4) is asymptotically for constants A,B as n→∞. This is proved by using a natural correspondence between partitions of integers and star-like trees.     

Stirling numbers and integer partitions

In this paper, we prove that the Stirling numbers of both kinds can be written as sums over integer partitions. As corollaries, we rewrite some identities with Stirling numbers of both kinds without Stirling numbers.     

Restricted partitions and q-Pell numbers

In this paper, we provide new combinatorial interpretations for the Pell numbers p _n in terms of finite set partitions. In particular, we identify six classes of partitions of size n, each avoiding a set of three classical patterns of length four, all of which have cardinality given by p _n . By restricting the statistic recording the number of inversions to one of these classes, and taking it jointly with the statistic recording the number of blocks, we obtain a new polynomial generalization of p _n . Similar considerations using the comajor index statistic yields a further generalization of the q-Pell number studied by Santos and Sills.     

Square-bounded partitions and Catalan numbers

For each integer k≥1, we define an algorithm which associates to a partition whose maximal value is at most k a certain subset of all partitions. In the case when we begin with a partition λ which is square-bounded, i.e. λ=(λ ₁≥⋅⋅⋅≥λ _k ) with λ ₁=k and λ _k =1, applying the algorithm ℓ times gives rise to a set whose cardinality is either the Catalan number c _ℓ−k+1 (the self dual case) or twice that Catalan number. The algorithm defines a tree and we study the propagation of the tree, which is not in the isomorphism class of the usual Catalan tree. The algorithm can also be modified to produce a two-parameter family of sets and the resulting cardinalities of the sets are the ballot numbers. Finally, we give a conjecture on the rank of a particular module for the ring of symmetric functions in 2ℓ+m variables.     

On mock theta functions and weight-attached Frobenius partitions

The Ramanujan Journal - In this paper, we provide the combinatorial interpretations of many mock theta functions and some generalizations using Frobenius partitions with attached weights. We...     

Catalan numbers,q-Catalan numbers and hypergeometric series

Catalan numbers are examined in the context of hypergeometric series. We are thus able to produce new and simple q-analogs related to the theory of partitions.     

Reverse plane partitions and tableau hook numbers

The generating function of R. P. Stanley for reverse plane partitions on a tableau shape is obtained by a direct method that clearly shows the combinatorial significance of the hook numbers for the shape. The process generalizes the hooks into zigzag paths.     

Bell numbers, determinants and series

In this article, we study Bell numbers and Uppuluri Carpenter numbers. We obtain various expressions and relations between them. These include polynomial recurrences and expressions as determinants of certain matrices of binomial coefficients.     

Newforms of squarefree level and theta series

We show that the cuspidal part of the span of the theta series associated to maximal integral lattices on a definite quaternion algebra over is precisely the space of newforms. Furthermore, using the Eichler commutation relation and an idea coming out of the Yoshida lifting, we show how to express each newform as an explicit linear combination of such theta series. The coefficients of these linear combinations come from cuspidal eigenvectors of Brandt matrices.     

Verlinde bundles and generalized theta linear series

In this paper we approach the study of generalized theta linear series on moduli of vector bundles on curves via vector bundle techniques on abelian varieties.

We study a naturally defined class of vector bundles on a Jacobian, called Verlinde bundles, in order to obtain information about duality between theta functions and effective global and normal generation on these moduli spaces.

     

Supersingular elliptic curves, theta series and weight two modular forms

Let be a prime, and let denote the space of weight two modular forms on all of whose Fourier coefficients are integral, except possibly for the constant term, which should be either integral or half-integral. We prove that is spanned as a -module by theta series attached to the unique quaternion algebra that is ramified at , at infinity, and at no other primes.

     

A brief history of partitions of numbers,partition functions and their modern applications

‘Number rules the universe.’

The Pythagoras     

Winding numbers and fourier series

This is an expository paper on a topic of classical analysis arising from the BMO-theory of topological degree (Brezis-Nirenberg, 1995). We sketch the history of the subject and some of its recent developments.     

Degenerate principal series and local theta correspondence II

Following our previous paper [LZ] which deals with the groupU(n, n), we study the structure of certain Howe quotients Ω ^p,q and Ω ^p,q (1) which are natural Sp(2n,R) modules arising from the Oscillator representation associated with the dual pair (O(p, q), Sp(2n,R)), by embedding them into the degenerate principal series representations of Sp(2n,R) studied in [L2].