Utilities to neaten permutation objects

Various utilities to neaten word objects by removing fixed elements

Usage

tidy(x)
trim(x)

Arguments

x: Object of class word, or in the case of tidy(), coerced to class word

Details

Function trim() takes a word and, starting from the right, strips off columns corresponding to fixed elements until it finds a non-fixed element. This makes no sense for cycle objects; if x is of class cycle, an error is returned.

Function tidy() is more aggressive. This firstly removes all fixed elements, then renames the non-fixed ones to match the new column numbers. The map is an isomorphism (sic) with respect to composition.

Value

Returns an object of class word

Author

Robin K. S. Hankin

Note

Results in empty (that is, zero-column) words if a vector of identity permutations is given

Examples

as.cycle(5:3)+as.cycle(7:9)
#> [1] (354)(789)
tidy(as.cycle(5:3)+as.cycle(7:9))
#> [1] (132)(456)
#> [coerced from word form]

as.cycle(tidy(c(as.cycle(1:2),as.cycle(6:7))))
#> [1] (12) (34)


nicify_cyclist(list(c(4,6), c(7), c(2,5,1), c(8,3)))
#> [[1]]
#> [1] 1 2 5
#> 
#> [[2]]
#> [1] 3 8
#> 
#> [[3]]
#> [1] 4 6
#> 

data(megaminx)
tidy(megaminx)  # has 120 columns, not 129
#>                                                                                          White 
#>                        (1,3,5,7,9)(2,4,6,8,10)(12,24,36,48,60)(13,25,37,49,51)(14,26,38,50,52) 
#>                                                                                         Purple 
#>                  (6,58,82,72,26)(7,59,83,73,27)(8,60,84,74,28)(11,13,15,17,19)(12,14,16,18,20) 
#>                                                                                     DarkYellow 
#>                 (8,20,80,70,38)(9,11,71,61,39)(10,12,72,62,40)(21,23,25,27,29)(22,24,26,28,30) 
#>                                                                                       DarkBlue 
#>              (1,23,69,109,41)(2,24,70,110,42)(10,22,68,108,50)(31,33,35,37,39)(32,34,36,38,40) 
#>                                                                                            Red 
#>               (2,34,106,96,52)(3,35,107,97,53)(4,36,108,98,54)(41,43,45,47,49)(42,44,46,48,50) 
#>                                                                                      DarkGreen 
#>                  (4,46,94,84,14)(5,47,95,85,15)(6,48,96,86,16)(51,53,55,57,59)(52,54,56,58,60) 
#>                                                                                     LightGreen 
#>         (21,79,111,101,31)(22,80,112,102,32)(30,78,120,110,40)(61,63,65,67,69)(62,64,66,68,70) 
#>                                                                                         Orange 
#>            (18,90,112,62,28)(19,81,113,63,29)(20,82,114,64,30)(71,73,75,77,79)(72,74,76,78,80) 
#>                                                                                      LightBlue 
#>            (16,56,92,114,74)(17,57,93,115,75)(18,58,94,116,76)(81,83,85,87,89)(82,84,86,88,90) 
#>                                                                                    LightYellow 
#>        (44,104,116,86,54)(45,105,117,87,55)(46,106,118,88,56)(91,93,95,97,99)(92,94,96,98,100) 
#>                                                                                           Pink 
#> (32,66,118,98,42)(33,67,119,99,43)(34,68,120,100,44)(101,103,105,107,109)(102,104,106,108,110) 
#>                                                                                           Grey 
#> (64,76,88,100,102)(65,77,89,91,103)(66,78,90,92,104)(111,113,115,117,119)(112,114,116,118,120) 
#> [coerced from word form]
stopifnot(all(unique(sort(unlist(as.cycle(tidy(megaminx)),recursive=TRUE)))==1:120))

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