A Young tableau is a finite collection of boxes, arranged in left-justified rows, with the row lengths in non-increasing order. A standard Young tableau is a labelled Young tableau whose the entries are increasing across each row and each column. We transform the boxes of Young tableau into triangle, establishing rules similar to standard Young tableau, and we name it triangular Young tableau.
In this research, we firstly derived the recursive relations, generating functions, and general terms for specific shapes of two-column triangular Young tableaux, which are related to well-known Catalan numbers in combinatorics. Afterward, we generalized this to arbitrary two-column triangular Young tableau, obtaining a formula to calculate their count. The logical and methodological approaches used in this research may be useful for future studies on more general(e.g. multi-column) triangular Young tableaux. Additionally, the research is related to partially ordered sets and might find applications in sorting problems in the field of information.