MS Thesis Defense: Menalu Mekcha

This thesis studies the enumerative combinatorics of skew and restricted Dyck path families. Beginning with classical Dyck paths, non-decreasing Dyck paths, and domino tilings, we review their connections to Catalan numbers and odd-indexed Fibonacci numbers. For partial skew Dyck paths, we use Prodinger’s decorated path framework and the kernel method to obtain level-by-level generating functions and a structural bijection explaining the level recurrence.

The main contribution concerns restricted skew Dyck paths whose valley heights form a non-decreasing sequence. By tracking semi-length and peak count, we derive the bivariate generating function S_UD(x,y)=xy(1-2x)/((1-2x)^2-xy(1-x)). Setting y=1 recovers the total enumeration as a Fibonacci binomial sum. Finally, extracting row polynomials from this generating function shows that the double-indexed array s_UD(n,m) gives a combinatorial realization of OEIS triangle A114164.