メモ帳 S-function の正規化

Littlewood の S-function の standardization

夏休みの勉強ｗ

分割の共役

　
の共役は

main = print (conj [4,3,3,2,2])

conj :: Partition -> Partition
conj [] = []
conj xs = conj' (xs ++ [0]) 

conj' :: Partition -> Partition
conj' [] = [] 
conj' [x] = [] 
conj' xs = (replicate (l2 - l1) p) ++ (conj' (init xs))
    where 
      p  = (length xs) - 1
      l1 = last xs
      l2 = last (init xs)

GHCi, version 8.0.1
   
[5,5,3,1]

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共役を取って

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分割｛λ｝の正規化

分割｛λ｝を S-function とみて、｛λ｝が弱い減少列になっていない時に弱い減少列に直す。

規則１　尻の 0 は取り去る。
規則２　尻に負数が出たら 0。（ここではヌルとする）
規則３　隣り合う数の入れ替えは=

行き当たりばったりで適当ｗ

main = print (standard (1, [2,1,3,6,1]))

type Partition = [Int]
type PartitionF = (Int, Partition) -- Partition with a (phase)factor

standard :: PartitionF -> PartitionF
standard (f, xs) = if stdq' xs then (f, xs)
                               else standard (std (f, xs))

std :: PartitionF -> PartitionF
std (f, [])= (f, [])
std (f, xs) = if (last xs) <  0 then (f, [])  
         else if (last xs) == 0 then std (f, init xs)
         else if (p1 xs)        then (f, [])
         else stdq (f, xs)

p1 :: Partition -> Bool
p1 xs = or [x == (y-1)|(x,y) <- ys ]
  where 
    ys = zip xs (tail xs)  
    
stdq :: PartitionF -> PartitionF
stdq (f, []) = (f, [])
stdq (f, xs) = if stdq' xs then (f, xs)
                           else std' (f, xs)
         
std' :: PartitionF -> PartitionF                           
std' (f, []) = (f, [])
std' (f, [x]) = (f, [x])
std' (f, x:y:xs) = if (x < y) then ex1 (f, x:y:xs)
                              else (f', x:xs')
    where 
      (f', xs') = std' (f, y:xs)

stdq' :: Partition -> Bool
stdq' xs = and [x >= y|(x,y) <- ys ]
  where 
    ys = zip xs (tail xs)     
    
ex1 :: PartitionF -> PartitionF
ex1 (f, x:y:xs) = (-f, (y-1):(x+1):xs) 

GHCi, version 8.0.1
   
(1,[3,3,3,3,1])