Weight (strings)

The $a$-weight of a string, for $a$ a letter, is the number of times that letter occurs in the string. More precisely, let $A$ be a finite set (called the "alphabet"), $ain\; A$ a "letter" of $A$, and $cin\; A^*$ a"string" (where $A^*$ is the free monoid generated by the elements of $A$, equivalently the set of strings, including the empty string, whose letters are from $A$). Then the $a$-"weight" of $c$, denoted by $mathrm\{wt\}\_a(c)$, is the number of times the generator $a$ occurs in the unique expression for $c$ as a product (concatenation) of letters in $A$.

If $A$ is an abelian group, the Hamming weight $mathrm\{wt\}(c)$ of $c$,often simply referred to as "weight", is the number of nonzero letters in $c$.

Examples

* Let $A=\{x,y,z\}$. In the string $c=yxxzyyzxyzzyx$, $y$ occurs 5times, so the $y$-weight of $c$ is $mathrm\{wt\}\_y(c)=5$.
* Let $A=mathbf\{Z\}\_3=\{0,1,2\}$ (an abelian group) and $c=002001200$. Then $mathrm\{wt\}\_0(c)=6$, $mathrm\{wt\}\_1(c)=1$, $mathrm\{wt\}\_2(c)=2$ and $mathrm\{wt\}(c)=mathrm\{wt\}\_1(c)+mathrm\{wt\}\_2(c)=3$.

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