order_of_ops.R
In permutations: The Symmetric Group: Permutations of a Finite Set

## ----label=setup,include=FALSE------------------------------------------------
library("permutations")
library("magrittr")

## ----label=definea------------------------------------------------------------
a <- as.cycle("(145)(26)")
a

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options("print_word_as_cycle" = FALSE)
(a <- as.word(a))

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a_active <- rbind(1:6,as.word(a))
rownames(a_active) <- c("place before move","place after move")
a_active

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inverse(a)

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a_passive <- rbind(1:6,as.word(inverse(a)))
rownames(a_passive) <- c("place after move","place before move")
a_passive

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b <- as.word(c(5, 2, 3, 4, 6, 1))
b

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a*b

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as.vector(b)[as.vector(a)]

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fa <- as.function(a)
fb <- as.function(b)
fb(fa(1))
as.function(a*b)(1)  # should match fb(fa(1))

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1 %>% fa %>% fb  # idiom for fb(fa(1)), should match result above

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a   # word form 
a_active  # matrix form (active)
a_passive  # matrix form (passive)

## ----echo=FALSE---------------------------------------------------------------
b_active <- rbind(1:6,as.word(b))
rownames(b_active) <- c("place before move","place after move")
b_passive <- rbind(1:6,as.word(inverse(b)))
rownames(b_passive) <- c("place after move","place before move")

## -----------------------------------------------------------------------------
b
b_active
b_passive

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a_passive
b_passive

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inverse(inverse(b) * inverse(a)) == a*b # both should be TRUE
inverse(b) * inverse(a) == inverse(a*b) # note b precedes a on LHS

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g <- as.cycle(c(1,2,6))
g

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pg <- perm_matrix(g)
pg

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dimnames(pg) <- list(place_before_move=1:6,place_after_move=1:6)
pg

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h <- as.word(c(1,3,4,5,2,6))
h
ph <- perm_matrix(h)
dimnames(ph) <- list(place_before_move=1:6,place_after_move=1:6)
ph

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pg %*% ph