FUNCTION: IsPerm - test whether the argument is a permutation

CALLING SEQUENCE:

IsPerm(expr)

TYP[IsPerm](expr)

PARAMETERS:

expr = any expression

SYNOPSIS:

• The IsPerm function tests whether expr is a permutation. A permutation of the symmetric group S(n) is a list of n distinct positive integers between 1 and n.

• One can test whether the permutation is either dominant, grassmannian or vexillary by using the corresponding second argument, 'dominant', 'grassmannian' or 'vexillary'.

• A permutation is dominant, if and only if its code is a partition (i.e. weakly decreasing vector of non-negative integers).

• A permutation is grassmannian, if and only if it has only one descent, or in other words, if there exists only one i with perm[i]>perm[i+1].

• Let part(perm) be the partition obtained by reordering the components of its code. Then perm is vexillary if and only if part(Perm2Inv(perm)) is the conjugate of part(perm).

• Whenever there is a conflict between the function name IsPerm and another name used in the same session, use the long form TYP['IsPerm'].

EXAMPLES:

> with(TYP):
> IsPerm([3,1]);
 
                                    false
 
> IsPerm([3,2,1]);
 
                                     true
 
> IsPerm([2,1,4,3], 'vexillary');
 
                                    false