We introduce the notion of doubly rooted plane trees and give a decomposition of these trees, called the butterfly decomposition which turns out to have many applications. From the butterfly decomposition we obtain a oneto-one correspondence between doubly rooted plane trees and free Dyck paths, which implies a simple derivation of a relation between the Catalan numbers and the central binomial...

Abstract This paper analyzes three formal models of Transformer encoders that differ in the form their self-attention mechanism: unique hard attention (UHAT); generalized (GUHAT), which generalizes UHAT; and averaging (AHAT). We show UHAT GUHAT Transformers, viewed as string acceptors, can only recognize languages complexity class AC0, recognizable by families Boolean circuits constant depth po...

In this paper, we give infinitely many examples of (non-isomorphic) connected k-regular graphs with smallest eigenvalue in half open interval [−1− √ 2,−2) and also infinitely many examples of (non-isomorphic) connected k-regular graphs with smallest eigenvalue in half open interval [α1,−1− √ 2) where α1 is the smallest root(≈ −2.4812) of the polynomial x3 + 2x2 − 2x − 2. From these results, we ...

iv Acknowledgements v List of Tables vii List of Graphs vii

................................................................................................................................................. 3 Acknowledgements................................................................................................................................ 6 List of Tables...........................................................................................

The number of down-steps between pairs up-steps in $k_t$-Dyck paths, a generalization Dyck paths consisting steps $\{(1, k), (1, -1)\}$ such that the path stays (weakly) above line $y=-t$, is studied. Results are proved bijectively and by means generating functions, lead to several interesting identities as well links other combinatorial structures. In particular, there connection perforation p...

A k-triangulation of a convex polygon is a maximal set of diagonals so that no k + 1 of them mutually cross in their interiors. We present a bijection between 2-triangulations of a convex n-gon and pairs of non-crossing Dyck paths of length 2(n−4). This gives a bijective proof of a recent result of Jonsson for the case k = 2. We obtain the bijection by constructing isomorphic generating trees f...