:: deftheorem Def5 defines Euler_transformation EULRPART:def 5 :
for n being Nat
for p being one-to-one a_partition of n
for b₃ being odd-valued a_partition of n holds
( b₃ = Euler_transformation p iff for j being Nat
for O being odd-valued FinSequence
for a being natural-valued FinSequence st len O = len p & len p = len a & p = O (#) (2 |^ a) holds
for sort being DoubleReorganization of dom p st 1 = O . (sort _ (1,1)) & ... & 1 = O . (sort _ (1,(len (sort . 1)))) & 3 = O . (sort _ (2,1)) & ... & 3 = O . (sort _ (2,(len (sort . 2)))) & 5 = O . (sort _ (3,1)) & ... & 5 = O . (sort _ (3,(len (sort . 3)))) & ( for i being Nat holds (2 * i) - 1 = O . (sort _ (i,1)) & ... & (2 * i) - 1 = O . (sort _ (i,(len (sort . i)))) ) holds
( card (Coim (b₃,1)) = ((2 |^ a) . (sort _ (1,1))) + (((((2 |^ a) *. sort) . 1),2) +...) & card (Coim (b₃,3)) = ((2 |^ a) . (sort _ (2,1))) + (((((2 |^ a) *. sort) . 2),2) +...) & card (Coim (b₃,5)) = ((2 |^ a) . (sort _ (3,1))) + (((((2 |^ a) *. sort) . 3),2) +...) & card (Coim (b₃,((j * 2) - 1))) = ((2 |^ a) . (sort _ (j,1))) + (((((2 |^ a) *. sort) . j),2) +...) ) );