We consider the construction of minimal multi-layered perceptrons for solving combinatorial optimization problems. Though general in nature, the construction method is presented as a case study for the sorting problem. The presentation starts with an 0((n!)2) three-layered perceptron based on complete enumeration, that solves the sorting problem of n numbers. This network is then gradually redu...

A recent progress on the complete enumeration of oriented matroids enables us to generate all combinatorial types of small point configurations and hyperplane arrangements in general dimension, including degenerate ones. This extends a number of former works which concentrated on the nondegenerate case and are usually limited to dimension 2 or 3. Our initial study on the complete list for small...

Concurrent systems are commonly veriied after computing a state graph describing all possible behaviors. Unfortunately, this state graph is often too large to be eeectively built. Partial-order techniques have been developed to avoid combinatorial explosion while preserving the properties of interest. This paper investigates the combination of two such approaches, persistent sets and covering s...

A law of large numbers and a central limit theorem are derived for linear statistics of random symmetric matrices whose on-or-above diagonal entries are independent, but neither necessarily identically distributed, nor necessarily all of the same variance. The derivation is based on systematic combinatorial enumeration, study of generating functions, and concentration inequalities of the Poinca...

A 2-binary tree is a binary rooted tree whose root is colored black and the other vertices are either black or white. We present several bijections concerning different types of 2-binary trees as well as other combinatorial structures such as ternary trees, non-crossing trees, Schröder paths, Motzkin paths and Dyck paths. We also obtain a number of enumeration results with respect to certain st...

We give a complete enumeration of combinatorial 3-manifolds with 10 vertices: There are precisely 247882 triangulated 3-spheres with 10 vertices as well as 518 vertex-minimal triangulations of the sphere product S×S and 615 triangulations of the twisted sphere product S×S. All the 3-spheres with up to 10 vertices are shellable, but there are 29 vertexminimal non-shellable 3-balls with 9 vertices.

We consider problems in sequence enumeration suggested by Stockhausen’s problem, and derive a generating series for the number of sequences of length k on n available symbols such that adjacent symbols are distinct, the terminal symbol occurs exactly r times, and all other symbols occur at most r − 1 times. The analysis makes extensive use of techniques from the theory of symmetric functions. E...

We consider two combinatorial statistics on permutations. One is the genus. The other, d̂es, is defined for alternating permutations, as the sum of the number of descents in the subwords formed by the peaks and the valleys. We investigate the distribution of d̂es on genus zero permutations and Baxter permutations. Our qenumerative results relate the d̂es statistic to lattice path enumeration, the ...

Throughout our study of the enumeration of plane partitions we make use of bijective proofs to find generating functions. In particular, we consider bounded plane partitions, symmetric plane partitions and weak reverse plane partitions. Using the combinatorial interpretations of Schur functions in relation to semistandard Young tableaux, we rely on the properties of symmetric functions. In our ...

The enumeration of connected graphs by excess (number of edges minus number of vertices) is a well-understood problem usually dealt by means of combinatorial decompositions or indirect formal series manipulations. In their work [3], Philippe Flajolet, Bruno Salvy, and Gilles Schaeffer derive such enumerative results mainly using analytic methods. In particular, they exhibit strong connections b...