an integer, the cardinality of the hook of that box, called the *hook length*. We give some illustrations in Figure \[fig.hooks\].

1

[@cc@c@]{} $\young({20}{19}{18}{17}{16}{15}{14}{12}{11}{10}987654321,8765431,654321)$ & $\young({10}98765,987654,876543,765432,654321)$ &

  --------------------------------
         $\young(65321,21)$
   $\young(421,1)$ $\young(31,1)$
  --------------------------------

Given a diagram $\lambda$, we can follow its *01-sequence* as the (bidirectional) sequence of steps taken when following the outline of the partition, starting at the bottom and ending to the right (see Figure \[fig.01\] for an example). A vertical step (up) is denoted by a 0, a horizontal step (right) by a 1. The 01-sequence $(x_i)$ of a partition, which is indexed by integers, is thus eventually 0 for negative $i$ and eventually 1 for positive $i$. We have only defined the sequence up to a shift so far, so we follow the additional convention that $|\{x_i =0: i \ge 0\}|= |\{x_i =1: i<  0\} |$. There is then a unique 01-sequence associated to each partition, and conversely this sequence determines the partition. We write $x(\lambda)$ for the 01-sequence of $\lambda$. Figure \[fig.sequence\] gives a particular example.

![The Young diagram of the partition $\lambda = (18,7,6)$ together with its 01-sequence $\cdots{\textcolor{blue}{0}}{\textcolor{red}{1}}{\textcolor{darkgreen}{1}}{\textcolor{blue}{1}}{{\textbf{\Large |}}}{\textcolor{red}{1}}{\textcolor{darkgreen}{1}}{\textcolor{blue}{1}}{\textcolor{red}{0}}{\textcolor{darkgreen}{1}}{\textcolor{blue}{0}}{\textcolor{red}{1}}{\textcolor{darkgreen}{1}}{\textcolor{blue}{1}}{\textcolor{red}{1}}{\textcolor{darkgreen}{1}}{\textcolor{blue}{1}}{\textcolor{red}{1}}{\textcolor{darkgreen}{1}}{\textcolor{blue}{1}}{\textcolor{red}{1}}{\textcolor{darkgreen}{1}}{\textcolor{blue}{0}}{\textcolor{red}{1}}\cdots$ (the ${{\textbf{\Large |}}}$ indicates the unique location where the number of 1s to the left 