Hook Length Formula For Skew Shapes And Enumeration On Certain Skew Standard Tableaux

Young tableaux are important structures in enumerative combinatorics.The famous hook length formula for counting standard Young tableaux was given in 1954.Enumeration formula for skew shapes was not known until 2014.In this paper,we first introduce Naruse's hook length formula for skew standard tableaux and a bijective proof of the formula.Naruse's hook length formula is a sum over excited diagrams of products of hook-lengths,instead of a closed formula.We give closed enumeration formulas by constructive method for certain skew standard tableaux.The results are as follows: the number of skew standard tableaux of shape ?/(1,1)is the Motzkin number;the number of skew standard tableaux of shape ?/? when |?| ? 2;and the number of skew standard tableaux of shape ?/? with a given descent when |?| ? 2.