Just as the Wolfram Language allows you to define how expressions should be evaluated, so also it allows you to define how expressions should be formatted for output. The basic idea is that whenever the Wolfram Language is given an expression to format for output, it first calls Format[expr] to find out whether any special rules for formatting the expression have been defined. By assigning a value to Format[expr] you can therefore tell the Wolfram Language that you want a particular kind of expression to be output in a special way.
This tells the Wolfram Language to format bin objects in a special way.
Now bin objects are output to look like binomial coefficients.
Internally, however, bin objects are still exactly the same.
give a definition only for a particular output form
Defining your own rules for formatting.
By making definitions for Format, you can tell the Wolfram Language to format a particular expression so as to look like another expression. You can also tell the Wolfram Language to run a program to determine how a particular expression should be formatted.
This specifies that the Wolfram Language should run a simple program to determine how xrep objects should be formatted.
The strings are created when each xrep is formatted.
Internally, however, the expression still contains xrep objects.
an object to be parenthesized with a precedence level n
Output forms for operators.
This prints with f represented by the "prefix operator" <>.
Here is output with the "infix operator" <>.
By default, the "infix operator" <> is assumed to have "higher precedence" than +, so no parentheses are inserted.
When you have an output form involving operators, the question arises of whether the arguments of some of them should be parenthesized. As discussed in "Special Ways to Input Expressions", this depends on the "precedence" of the operators. When you set up output forms involving operators, you can use PrecedenceForm to specify the precedence to assign to each operator. The Wolfram Language uses integers from 1 to 1000 to represent "precedence levels". The higher the precedence level for an operator, the less it needs to be parenthesized.
Here is treated as an operator with precedence 100. This precedence turns out to be low enough that parentheses are inserted.
When you make an assignment for Format[expr], you are defining the output format for expr in all standard types of Wolfram Language output. By making definitions for Format[expr,form], you can specify formats to be used in specific output forms.