Barred Permutations
These partitions are elements of the wreath product S2 wr Sn, the
Hyperoctahedral group. These elements are
objects of the type PERMUTATION, but now the kind of the
permutation is BAR or BARCYCLE.
The permutation is of length n, and the
entries are between 1 and n, but here they can be positive or negative.
The first routine we describe is that which
computes the label of the conjugacy class of the
element a.
- NAME: class_bar
- SYNOPSIS: INT class_bar(OP a,b)
In order to use the next routine,
you enter the label of a class, and the routine computes
an representative element b of that class:
- NAME: class_rep_bar
- SYNOPSIS: INT class_rep_bar(OP a,b)
Another routine
computes the Lehmer code of a barred permutation a.
The result is a two-element VECTORobject, whose two entries
are INTEGER - VECTOR objects, whose lengths are the length of the
PERMUTATIONobject a.
The first VECTOR is a 0-1-vector, the i-th entry is one if the
element i+1 is negative in the PERMUTATIONobject a.
The second VECTOR is the ordinary Lehmer code of a permutation,
but taken into account that we may have negative entries.
- NAME: lehmercode_bar
- SYNOPSIS: INT lehmercode_bar(OP a,b)
- An EXAMPLE: If we are given the barred permutation
3 -5 2 -1 4
the result of an application of lehmercode_bar is
[[1,0,0,0,1][3,0,1,0,0]]
The next routine
is the inverse routine of the above routine:
- NAME: lehmercode_vector_bar
- SYNOPSIS: INT lehmercode_vector_bar(OP a,b)
The following routine
computes the reduced length of the barred permutation a.
- NAME: length_bar
- SYNOPSIS: INT length_bar(OP a,b)
This routine
computes a vector containing all the labelings of the classes of
the group S2 wr Sa:
- NAME: makevectorof_class_bar
- SYNOPSIS: INT makevectorof_class_bar(OP a,b)
Another routine
computes a vector with representatives of all the conjugacy classes of
the group S2 wr Sa, the ordering of classes is as in the
function makevectorof_class_bar.
- NAME: makevectorof_class_rep_bar
- SYNOPSIS: INT makevectorof_class_rep_bar(OP a,b)
The following routine
computes the order of the centraliser of the class
labeled by a.
- NAME: ordcen_bar
- SYNOPSIS: INT ordcen_bar(OP a,b)
This routine
multiplies a and b (as permutations: ''first b then a):
- NAME: mult_bar_bar
- SYNOPSIS: INT mult_bar_bar(OP a,b,c)
This one
computes the order of the class labeled by a.
- NAME: ordcon_bar
- SYNOPSIS: INT ordcon_bar(OP a,b)
The next routine
computes a random element of given length, so b becomes
an element of S2 wr Sa, while a is an INTEGERobject:
- NAME: random_bar
- SYNOPSIS: INT random_bar(OP a,b)
Here is a routine that
transforms a barred permutation a in list-notation into
cycle notation
- NAME: t_BAR_BARCYCLE
- SYNOPSIS: INT t_BAR_BARCYCLE(OP a,b)
- BUG: a and b must be different
The next routine
transforms a barred permutation a in cycle-notation into
list notation:
- NAME: t_BARCYCLE_BAR
- SYNOPSIS: INT t_BARCYCLE_BAR(OP a,b)
- BUG: a and b must be different
harald.fripertinger@kfunigraz.ac.at,
last changed: November 19, 2001
