CLASSIFICATION OF STANDARD YOUNG TABLEAUX IN A ( 3 , 0 ) $(3,0)$left parenthesis 3 comma 0 right parenthesis- AND ( 2 , 1 ) $(2,1)$left parenthesis 2 comma 1 right parenthesis-HOOK

We explore the combinatorial properties of the set S ( k , ℓ ; n ) S(k,; n) German upper S left parenthesis k comma script l semicolon n right parenthesis consisting of standard Young tableaux with specific shape constraints. Using bijections with Motzkin and free Motzkin paths, we enumerate S m ( 3 , 0 ; n ) Sₘ(3,0;n) normal upper S Subscript m Baseline left parenthesis 3 comma 0 semicolon n right parenthesis and S m ( 2 , 1 ; n ) Sₘ(2,1;n) normal upper S Subscript m Baseline left parenthesis 2 comma 1 semicolon n right parenthesis for all m , and S m ( 0 , 3 ; n ) Sₘ(0,3;n) normal upper S Subscript m Baseline left parenthesis 0 comma 3 semicolon n right parenthesis and S m ( 1 , 2 ; n ) Sₘ(1,2;n) normal upper S Subscript m Baseline left parenthesis 1 comma 2 semicolon n right parenthesis for small values of m . We derive explicit formulae such as S 2 ( 3 , 0 ; n ) = <mml