Shape of a permutation

Returns the shape of a permutation. If given a word, it coerces to cycle form.

Usage

shape(x, drop = TRUE,id1 = TRUE,decreasing = FALSE)
shape_cyclist(cyc,id1=TRUE)
padshape(x, drop = TRUE, n=NULL)
shapepart(x)
shapepart_cyclist(cyc,n=NULL)

Arguments

x: Object of class cycle (if not, coerced)
cyc: A cyclist
n: Integer governing the size of the partition assumed, with default NULL meaning to use the largest element
drop: Boolean, with default TRUE meaning to unlist if possible
id1: Boolean, with default TRUE in function shape_cyclist() meaning that the shape of the identity is “1” and FALSE meaning that the shape is NULL
decreasing: In function shape(), Boolean with default FALSE meaning to return the cycle lengths in the order given by nicify_cyclist() and TRUE meaning to sort in decreasing order

Value

Function shape() returns a list with elements representing the lengths of the component cycles.

Function shapepart() returns an object of class partition showing the permutation as a set partition of disjoint cycles.

Function padshape() returns a list of shapes but padded with ones so the total is the size of the permutation.

shapepart_cyclist() and shapepart_cyclist() are low-level helper functions.

Author

Robin K. S. Hankin

Note

What I call “shape”, others call the “cycle type”, so you will sometimes see “cycle type determines conjugacy class” as a theorem.

Examples


jj <- as.cycle(c("123","","(12)(34)","12345"))
jj
#> [1] (123)    ()       (12)(34) (12345) 
shape(jj)
#> [[1]]
#> [1] 3
#> 
#> [[2]]
#> [1] 1
#> 
#> [[3]]
#> [1] 2 2
#> 
#> [[4]]
#> [1] 5
#> 

shape(rperm(10,9)) # coerced to cycle
#> [[1]]
#> [1] 6 3
#> 
#> [[2]]
#> [1] 6 3
#> 
#> [[3]]
#> [1] 4 5
#> 
#> [[4]]
#> [1] 6 2
#> 
#> [[5]]
#> [1] 6 3
#> 
#> [[6]]
#> [1] 4 2
#> 
#> [[7]]
#> [1] 2 4 2
#> 
#> [[8]]
#> [1] 9
#> 
#> [[9]]
#> [1] 7 2
#> 
#> [[10]]
#> [1] 3 5
#> 

a <- rperm()
identical(shape(a,dec=TRUE),shape(a^cyc_len(2),dec=TRUE))
#> [1] TRUE


data(megaminx)
shape(megaminx)
#> $White
#> [1] 5 5 5 5 5
#> 
#> $Purple
#> [1] 5 5 5 5 5
#> 
#> $DarkYellow
#> [1] 5 5 5 5 5
#> 
#> $DarkBlue
#> [1] 5 5 5 5 5
#> 
#> $Red
#> [1] 5 5 5 5 5
#> 
#> $DarkGreen
#> [1] 5 5 5 5 5
#> 
#> $LightGreen
#> [1] 5 5 5 5 5
#> 
#> $Orange
#> [1] 5 5 5 5 5
#> 
#> $LightBlue
#> [1] 5 5 5 5 5
#> 
#> $LightYellow
#> [1] 5 5 5 5 5
#> 
#> $Pink
#> [1] 5 5 5 5 5
#> 
#> $Grey
#> [1] 5 5 5 5 5
#> 

jj <- megaminx*megaminx[1]
identical(shape(jj),shape(tidy(jj)))  #tidy() does not change shape
#> [1] TRUE


allperms(3)
#> [1] ()    (23)  (12)  (123) (132) (13) 
#> [coerced from word form]
shapepart(allperms(3))
#>                 
#> [1,] 1 2 1 1 1 1
#> [2,] 2 1 1 1 1 2
#> [3,] 3 1 2 1 1 1
shapepart(rperm(10,5))
#>                         
#> [1,] 1 1 3 1 2 1 1 1 1 1
#> [2,] 2 1 1 1 1 1 1 2 1 1
#> [3,] 2 1 1 1 1 1 1 1 2 2
#> [4,] 1 2 2 1 1 1 1 1 1 3
#> [5,] 2 1 2 1 1 1 1 1 1 2

shape_cyclist(list(1:4,8:9))
#> [1] 4 2
shapepart_cyclist(list(1:4,8:9))
#> [1] 1 1 1 1 3 4 5 2 2