Three enumeration formulas of standard Young tableaux of truncated shapes

In this paper we consider the enumeration of three kinds of standard Young tableaux (SYT) of truncated shapes by use of the method of multiple integrals. A product formula for the number of truncated shapes of the form (n^m, n ? r)δ_k–1 is given, which implies that the number of SYT of truncated shape (n², 1)(1) is the number of level steps in all 2-Motzkin paths. The number of SYT with three rows truncated by some boxes ((n + k)³)(k) is discussed. Furthermore, the integral representation of the number of SYT of truncated shape (n^m)(3, 2) is derived, which implies a simple formula of the number of SYT of truncated shape (nⁿ)(3, 2).