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The generalized weighted probability measure on the symmetric group and the asymptotic behavior of the cycles

Ashkan Nikeghbali, Dirk Zeindler (2013)

Annales de l'I.H.P. Probabilités et statistiques

The goal of this paper is to analyse the asymptotic behaviour of the cycle process and the total number of cycles of weighted and generalized weighted random permutations which are relevant models in physics and which extend the Ewens measure. We combine tools from combinatorics and complex analysis (e.g. singularity analysis of generating functions) to prove that under some analytic conditions (on relevant generating functions) the cycle process converges to a vector of independent Poisson variables...

The number of complete exceptional sequences for a Dynkin algebra

Mustafa Obaid, Khalid Nauman, Wafa S. M. Al-Shammakh, Wafaa Fakieh, Claus Michael Ringel (2013)

Colloquium Mathematicae

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 operation and * operation of Cohen-Macaulay bipartite graphs

Yulong Yang, Guangjun Zhu, Yijun Cui, Shiya Duan (2024)

Czechoslovak Mathematical Journal

Let G be a finite simple graph with the vertex set V and let I G be its edge ideal in the polynomial ring S = 𝕂 [ V ] . We compute the depth and the Castelnuovo-Mumford regularity of S / I G when G = G 1 G 2 or G = G 1 * G 2 is a graph obtained from Cohen-Macaulay bipartite graphs G 1 , G 2 by the operation or * operation, respectively.

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