conjugate | R Documentation |

Given a partition, provide its conjugate or Durfee square

conjugate(x, sorted = TRUE) durfee(x, sorted = TRUE) durfee_sorted(x)

`x` |
Either a vector describing a partition or a matrix whose columns are partitions. |

`sorted` |
A logical indicating whether the data is already in standard form. That is to say, are the data within each colum sorted in decreasing order? |

Conjugation is described in Andrews, and (eg) Hardy and Wright.

The conjugate of a partition may be calculated by taking its Ferrers diagram and considering the partition defined by columns instead of rows. This may be visualised by flipping the Ferrers diagram about the leading diagonal.

Essentially, `conjugate()`

carries out **R** idiom

`rev(cumsum(table(factor(a[a>0],levels=max(a):1))))`

but is faster.

The “Durfee square” of a partition is defined on page 281 of
Hardy and Wright. It is the largest square of nodes contained in the
partition's Ferrers graph. Function `durfee()`

returns the
length of the side
of the Durfee square, which Andrews denotes
*d(lambda)*. It is equivalent to **R** idiom

`function(a){sum(a>=1:length(a))}`

but is faster.

Returns either a partition in standard form, or a matrix whose columns are partitions in standard form.

If argument `x`

is not non-increasing, you must use the
`sorted = FALSE`

flag. Otherwise, these functions will not work and will
silently return garbage. Caveat emptor! (output from `blockparts()`

is not necessarily non-increasing)

Robin K. S. Hankin

parts(5) conjugate(parts(5)) restrictedparts(6,4) conjugate(restrictedparts(6,4)) durfee(10:1) # A parition in nonstandard form --- use `sorted = FALSE` x <- parts(5)[sample(5),] durfee(x, sorted = FALSE) conjugate(x, sorted = FALSE) # Suppose one wanted partitions of 8 with no part larger than 3: conjugate(restrictedparts(8,3)) # (restrictedparts(8,3) splits 8 into at most 3 parts; # so no part of the conjugate partition is larger than 3).

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