Another SPT crank for the number of smallest parts in overpartitions with even smallest part

Graphs with smallest forgotten index

The forgotten topological index of a molecular graph $G$ is defined as $F(G)=sum_{vin V(G)}d^{3}(v)$, where $d(u)$ denotes the degree of vertex $u$ in $G$. The first through the sixth smallest forgotten indices among all trees, the first through the third smallest forgotten indices among all connected graph with cyclomatic number $gamma=1,2$, the first through<br /...

`-adic Properties of Smallest Parts Functions

We prove explicit congruences modulo powers of arbitrary primes for three smallest parts functions: one for partitions, one for overpartitions, and one for partitions without repeated odd parts. The proofs depend on `-adic properties of certain modular forms and mock modular forms of weight 3/2 with respect to the Hecke operators T (`).

Multiplicity and Number of Parts in Overpartitions

The purpose of this paper is to study the parts, part sizes and multiplicities in overpartitions using combinatorics, probabilities and asymptotics. We show that the probability that a randomly chosen part size of a randomly chosen overpartition of n has multiplicity m or m + 1 approaches 1/(m(m+1) ln 2) and that the expected multiplicity of a randomly chosen part size of a randomly chosen over...

Constructing regular graphs with smallest defining number

In a given graph G, a set S of vertices with an assignment of colors is a defining set of the vertex coloring of G, if there exists a unique extension of the colors of S to a χ(G)-coloring of the vertices of G. A defining set with minimum cardinality is called a smallest defining set (of vertex coloring) and its cardinality, the defining number, is denoted by d(G, χ). Let d(n, r, χ = k) be the ...

Selecting the Smallest Extreme Value Population with the Smallest Variance