Discrete and Linear Structures in Enumeration

Abstract

The central theme of this thesis is to nd the multiset version of the combinatorial identities arising from the cyclic decomposition of permutations of nite sets. The main contributions of author's work are as follows. In Chapter 1, we nd the appropriate multiset version of the Stirling cycle number. Then, using these new Stirling numbers, we give a new equivalent statement of the coin arrangements lemma which is an important trick in Sherman's proof of Feynman conjecture on two dimensional Ising model. We also present a new proof of the coin arrangements lemma. Finally, we show several relations of the coin arrangements lemma with various concepts in enumerative combinatorics. In Chapter 2, we rst give a new proof of the Witt identity which is an algebraic identity in the context of Lyndon words using the Bass' identity for zeta function of nite graphs. Then, we present a new proof of the Bass' identity by only slight modi cations to the approach that has been developed by Feynman and Sherman as the path method for combinatorial solution of two dimensional Ising problem. In Chapter 3, we give a multiset generalization of the well-known graph-theoretical interpretation of the determinant as a signed weighted sum over cycle covers. In Chapter 4, we nd a multiset generalization of the graph-theoretical...