Rearrange the digits in ⟨125034⟩ to meet the rules below.
⟨ ⁵ᵗʰ▨ ⁴ᵗʰ▨ ³ʳᵈ▨ ²ⁿᵈ▨ ¹ˢᵗ▨ ⁰ᵗʰ▨ ⟩
✅Match
⟦1,2⟧ ∋ 3,4
⟦0,3⟧ ∋ 4,5
⟦2,4⟧ ∋ 5
⛔Avoid
⟨⋯ ᵃb ⋯⟩, |a-b|=0
⟨⋯ ᵃb ⋯⟩, |a-b|=5
⟨⋯ ᵃb ⋯⟩, |a-b|=4
----- Information -----
Your goal is to find a permutation that satisfies every required pattern ✅ and avoids every forbidden pattern ⛔ simultaneously.
There are 6 positions labeled 5th down to 0th, arranged left to right in descending order (5th is leftmost, 0th is rightmost). Therefore, ⟨125034⟩ = ⟨ ⁵ᵗʰ1 ⁴ᵗʰ2 ³ʳᵈ5 ²ⁿᵈ0 ¹ˢᵗ3 ⁰ᵗʰ4 ⟩.
“⋯” means there may be zero or more digits there. If digit A is in the N-th position, then we write ⟨⋯ ᴺA ⋯⟩.
----- Required patterns -----
✅ 「⟦1,2⟧ ∋ 3,4」
Digits 3,4 are positioned between digits 1 and 2.
There are 120 permutations matching this pattern.
Examples: ⟨253041⟩, ⟨153420⟩, ⟨103542⟩, ⟨143250⟩.
✅ 「⟦0,3⟧ ∋ 4,5」
Digits 4,5 are positioned between digits 0 and 3.
There are 120 permutations matching this pattern.
Examples: ⟨354201⟩, ⟨205431⟩, ⟨315240⟩, ⟨102543⟩.
✅ 「⟦2,4⟧ ∋ 5」
Digit 5 is positioned between digits 2 and 4.
There are 240 permutations matching this pattern.
Examples: ⟨345201⟩, ⟨452310⟩, ⟨231504⟩, ⟨123540⟩.
----- Forbidden patterns -----
⛔ 「⟨⋯ ᵃb ⋯⟩, |a-b|=0」
There exist a,b such that [a-th]=b, |a-b|=0.
There are 265 permutations avoiding this pattern.
Examples: ⟨215304⟩, ⟨031524⟩, ⟨132054⟩, ⟨312045⟩.
⛔ 「⟨⋯ ᵃb ⋯⟩, |a-b|=5」
There exist a,b such that [a-th]=b, |a-b|=5.
There are 504 permutations avoiding this pattern.
Examples: ⟨541302⟩, ⟨243510⟩, ⟨350214⟩, ⟨502341⟩.
⛔ 「⟨⋯ ᵃb ⋯⟩, |a-b|=4」
There exist a,b such that [a-th]=b, |a-b|=4.
There are 362 permutations avoiding this pattern.
Examples: ⟨052413⟩, ⟨425013⟩, ⟨032415⟩, ⟨415230⟩.
#125034_v2.11
