Note on a partition function

A Note on Tree-partition-width

A tree-partition of a graph G is a proper partition of its vertex set into ‘bags’, such that identifying the vertices in each bag produces a forest. The treepartition-width of G is the minimum number of vertices in a bag in a tree-partition of G. An anonymous referee of the paper by Ding and Oporowski [J. Graph Theory, 1995] proved that every graph with tree-width k ≥ 3 and maximum degree ∆ ≥ 1...

A Note on Strong Negative Partition Relations

We analyze a natural function definable from a scale at a singular cardinal, and provide a fairly simple proof that there is a strong connection between failures of square brackets partition relations at sucessors of singular cardinals and the club–guessing ideal idp(C̄, Ī).

A Note on the Partition Function of ABJM Theory on S 3 231

On a Multiplicative Partition Function

Let D(s) = ∑∞ m=1 amm −s be the Dirichlet series generated by the infinite product ∏∞ k=2(1− k−s). The value of am for squarefree integers m with n prime factors depends only on the number n, and we let f(n) denote this value. We prove an asymptotic estimate for f(n) which allows us to solve several problems raised in a recent paper by M. V. Subbarao and A. Verma.

On Stanley's Partition Function

Stanley defined a partition function t(n) as the number of partitions λ of n such that the number of odd parts of λ is congruent to the number of odd parts of the conjugate partition λ modulo 4. We show that t(n) equals the number of partitions of n with an even number of hooks of even length. We derive a closed-form formula for the generating function for the numbers p(n)− t(n). As a consequen...