The monoid of ordered partitions of a natural number.
The monotone cumulants
In the present paper we define the notion of generalized cumulants which gives a universal framework for commutative, free, Boolean and especially, monotone probability theories. The uniqueness of generalized cumulants holds for each independence, and hence, generalized cumulants are equal to the usual cumulants in the commutative, free and Boolean cases. The way we define (generalized) cumulants needs neither partition lattices nor generating functions and then will give a new viewpoint to cumulants....
The Nekrasov-Okounkov hook length formula: refinement, elementary proof, extension and applications
The paper is devoted to the derivation of the expansion formula for the powers of the Euler Product in terms of partition hook lengths, discovered by Nekrasov and Okounkov in their study of the Seiberg-Witten Theory. We provide a refinement based on a new property of -cores, and give an elementary proof by using the Macdonald identities. We also obtain an extension by adding two more parameters, which appears to be a discrete interpolation between the Macdonald identities and the generating function...
The number of 0-1-2 increasing trees as two different evaluations of the Tutte polynomial of a complete graph.
The number of 1-factors in some polyhexes.
The Number Of Antichains Of Finite Power Sets
The number of antichains of finite power sets.
The number of binomial coefficients in residue classes modulo p and .
The number of certain -combinations of an -set.
The number of complete exceptional sequences for a Dynkin algebra
The Dynkin algebras are the hereditary artin algebras of finite representation type. The paper determines the number of complete exceptional sequences for any Dynkin algebra. Since the complete exceptional sequences for a Dynkin algebra of Dynkin type Δ correspond bijectively to the maximal chains in the lattice of non-crossing partitions of type Δ, the calculations presented here may also be considered as a categorification of the corresponding result for non-crossing partitions.
The number of crossings in a regular drawing of the complete bipartite graph.
The number of elements in the mutation class of a quiver of type .
The number of -matchings in almost every tree is a zero residue.
The number of graphs not containing as a minor.
The number of indecomposable Schur rings over a cyclic 2-group.
The number of inequivalent optimal covering codes.
The number of inversions in permutations: A saddle point approach.
The number of knight's tours equals 33, 439, 123, 484, 294—counting with binary decision diagrams.
The number of labeled 2-connected planar graphs.
The number of [old-time] basketball games with final score : where the home team was never losing but also never ahead by more than points.