page_title: Hook length formula
section_title: Examples
hierarchical_section_title: Hook length formula / Examples
caption_attribution_description: English: A hook-length tableau of shape (4, 3, 1, 1), English notation
context_page_description: In combinatorial mathematics, the hook length formula is a formula for the number of standard Young tableaux whose shape is a given Young diagram.
  It has applications in diverse areas such as representation theory, probability, and algorithm analysis; for example, the problem of longest increasing subsequences. A related formula counts the number of semi-standard Young tableaux, which is a specialization of a Schur polynomial.
context_section_description: The figure on the right shows hook lengths for the cells in the Young diagram , corresponding to the partition 9 = 4 + 3 + 1 + 1. The hook length formula gives the number of standard Young tableaux as:
  A Catalan number counts Dyck paths with  steps going up (U) interspersed with  steps going down (D), such that at each step there are never more preceding D's than U's. These are in bijection with the Young tableaux of shape : a Dyck path corresponds to the tableau whose first row lists the positions of the U-steps, while the second row lists the positions of the D-steps. For example, UUDDUD correspond to the tableaux with rows 125 and 346.
  This shows that , so the hook formula specializes to the well-known product formula
caption: A tableau listing the hook length of each cell in the Young diagram 
   
    
      
        
          (
          4
          ,
          3
          ,
          1
          ,
          1
          )
        
      
      {\displaystyle (4,3,1,1)}
exif: {}