Related papers: Notes On U(1) Instanton Counting On $A_{l-1}$ ALE …
There have been two distinct schemes studied in the literature for instanton counting in A_{p-1} asymptotically locally Euclidean (ALE) spaces. We point out that the two schemes---namely the counting of orbifolded instantons and instanton…
This note introduces some bijections relating core partitions and tuples of integers. We apply these bijections to count the number of cores with various types of restriction, including fixed number of parts, limited size of parts, parts…
In 2003, Maroti showed that one could use the machinery of l-cores and l-quotients of partitions to establish lower bounds for p(n), the number of partitions of n. In this paper we explore these ideas in the case l=2, using them to give a…
We study a correspondence between numerical sets and integer partitions that leads to a bijection between simultaneous core partitions and the integer points of a certain polytope. We use this correspondence to prove combinatorial results…
An n-core partition is an integer partition whose Young diagram contains no hook lengths equal to n. We consider partitions that are simultaneously a-core and b-core for two relatively prime integers a and b. These are related to abacus…
We consider N=4 theories on ALE spaces of $A_{k-1}$ type. As is well known, their partition functions coincide with $A_{k-1}$ affine characters. We show that these partition functions are equal to the generating functions of some peculiar…
Johnson recently proved Armstrong's conjecture which states that the average size of an $(a,b)$-core partition is $(a+b+1)(a-1)(b-1)/24$. He used various coordinate changes and one-to-one correspondences that are useful for counting…
We survey some features of equivariant instanton partition functions of topological gauge theories on four and six dimensional toric Kahler varieties, and their geometric and algebraic counterparts in the enumerative problem of counting…
We give a bijection between the set of self-conjugate partitions and that of ordinary partitions. Also, we show the relation between hook lengths of self conjugate partition and corresponding partition via the bijection. As a corollary, we…
In this paper, we are mainly concerned with the enumeration of $(2k+1, 2k+3)$-core partitions with distinct parts. We derive the number and the largest size of such partitions, confirming two conjectures posed by Straub.
Simultaneous core partitions have attracted much attention since Anderson's work on the number of $(t_1,t_2)$-core partitions. In this paper we focus on simultaneous core partitions with distinct parts. The generating function of $t$-core…
This is the 13th article in the collection of reviews "Exact results in N=2 supersymmetric gauge theories", ed. J. Teschner. It discusses the relation between the instanton partition functions and the partition function of the topological…
In this paper we compute the partition function of 5D supersymmetric U(1) gauge theory with extra adjoint matter in general $\Omega$-background. It is well known that such partition functions encode very rich topological information. We…
We derive the precise relation between level matching condition and fractional instanton numbers in six dimensional, abelian and supersymmetric orbifolds of E8 x E8 heterotic string theory. The fractional part of the two E8 instanton…
This is the third article in the collection of reviews "Exact results on N=2 supersymmetric gauge theories", ed. J. Teschner. It is explained how to compute the instanton partition functions. The results can be written as sums over bases…
A special case of an elegant result due to Anderson proves that the number of $(s,s+1)$-core partitions is finite and is given by the Catalan number $C_s$. Amdeberhan recently conjectured that the number of $(s,s+1)$-core partitions into…
We study instanton partition functions for N=2 superconformal Sp(1) and SO(4) gauge theories. We find that they agree with the corresponding U(2) instanton partitions functions only after a non-trivial mapping of the microscopic gauge…
In this expository paper we summarize the connection between random partitions and 4-dimensional supersymmetric gauge theories discovered in hep-th/0306238. There is also a brief section about how this extends to 3-dimensional partitions…
In this paper, we study $(s,s+1)$-core partitions with $d$-distinct parts. We obtain results on the number and the largest size of such partitions, so we extend Xiong's paper in which the results are obtained about $(s,s+1)$-core partitions…
We answer a question of Zeilberger and Zeilberger about certain partition statistics.