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r tr strtr rrts t tr rs stt
❙♦❜♦❧❡✈ ■♥st✐t✉t❡ ♦❢ ▼❛t❤❡♠❛t✐❝s✱ ◆♦✈♦s✐❜✐rs❦✱ ❘✉ss✐❛ ◆♦✈♦s✐❜✐rs❦ ❙t❛t❡ ❯♥✐✈❡rs✐t②✱ ◆♦✈♦s✐❜✐rs❦✱ ❘✉ss✐❛ ❈♦♠✐♥❛t♦r✐❝s ❙❡♠✐♥❛r ❙❏❚❯ ✾ ❆♣r✐❧ ✷✵✶✽✱ ❙❤❛♥❣❤❛✐
❚❤❡ ❙t❛r ❣r❛♣❤ Sn = Cay(Symn, T), n 2
✐s t❤❡ ❈❛②❧❡② ❣r❛♣❤ ♦♥ t❤❡ s②♠♠❡tr✐❝ ❣r♦✉♣ Symn ♦❢ ♣❡r♠✉t❛t✐♦♥s π = [π1π2...πi...πn] ✇✐t❤ t❤❡ ❣❡♥❡r❛t✐♥❣ s❡t T ♦❢ ❛❧❧ tr❛♥s♣♦s✐t✐♦♥s ti = (1 i) s✇❛♣♣✐♥❣ t❤❡ 1st ❛♥❞ it❤ ❡❧❡♠❡♥ts ♦❢ ❛ ♣❡r♠✉t❛t✐♦♥ π✳
Pr♦♣❡rt✐❡s ♦❢ t❤❡ ❙t❛r ❣r❛♣❤
❝♦♥♥❡❝t❡❞ ❜✐♣❛rt✐t❡ ✕r❡❣✉❧❛r ❣r❛♣❤ ♦❢ ♦r❞❡r ❛♥❞ ❞✐❛♠❡t❡r ✭❙✳ ❇✳ ❆❦❡rs✱ ❇✳ ❑r✐s❤♥❛♠✉rt❤②✱ ✮ ✈❡rt❡①✲tr❛♥s✐t✐✈❡ ❛♥❞ ❡❞❣❡✲tr❛♥s✐t✐✈❡ ❝♦♥t❛✐♥s ❤❛♠✐❧t♦♥✐❛♥ ❝②❝❧❡s ✭❱✳ ❑♦♠♣❡❧✬♠❛❦❤❡r✱ ❱✳ ▲✐s❦♦✈❡ts✱ ✱ P✳ ❙❧❛t❡r✱ ✮ ✐t ❞♦❡s ❝♦♥t❛✐♥ ❡✈❡♥ ✕❝②❝❧❡s ✇❤❡r❡ ❤❛s ❤✐❡r❛r❝❤✐❝❛❧ str✉❝t✉r❡ ❤❛s ✐♥t❡❣r❛❧ s♣❡❝tr✉♠
❊❧❡♥❛ ❑♦♥st❛♥t✐♥♦✈❛ Pr♦♣❡rt✐❡s ♦❢ t❤❡ ❙t❛r ❣r❛♣❤s ❙❏❚❯✲✷✵✶✽ ✷ ✴ ✸✺
❚❤❡ ❙t❛r ❣r❛♣❤ Sn = Cay(Symn, T), n 2
✐s t❤❡ ❈❛②❧❡② ❣r❛♣❤ ♦♥ t❤❡ s②♠♠❡tr✐❝ ❣r♦✉♣ Symn ♦❢ ♣❡r♠✉t❛t✐♦♥s π = [π1π2...πi...πn] ✇✐t❤ t❤❡ ❣❡♥❡r❛t✐♥❣ s❡t T ♦❢ ❛❧❧ tr❛♥s♣♦s✐t✐♦♥s ti = (1 i) s✇❛♣♣✐♥❣ t❤❡ 1st ❛♥❞ it❤ ❡❧❡♠❡♥ts ♦❢ ❛ ♣❡r♠✉t❛t✐♦♥ π✳
Pr♦♣❡rt✐❡s ♦❢ t❤❡ ❙t❛r ❣r❛♣❤
❝♦♥♥❡❝t❡❞ ❜✐♣❛rt✐t❡ (n − 1)✕r❡❣✉❧❛r ❣r❛♣❤ ♦❢ ♦r❞❡r n! ❛♥❞ ❞✐❛♠❡t❡r diam(Sn) = ⌊ 3(n−1)
2
⌋ ✭❙✳ ❇✳ ❆❦❡rs✱ ❇✳ ❑r✐s❤♥❛♠✉rt❤②✱ 1989✮ ✈❡rt❡①✲tr❛♥s✐t✐✈❡ ❛♥❞ ❡❞❣❡✲tr❛♥s✐t✐✈❡ ❝♦♥t❛✐♥s ❤❛♠✐❧t♦♥✐❛♥ ❝②❝❧❡s ✭❱✳ ❑♦♠♣❡❧✬♠❛❦❤❡r✱ ❱✳ ▲✐s❦♦✈❡ts✱ 1975✱ P✳ ❙❧❛t❡r✱ 1978✮ ✐t ❞♦❡s ❝♦♥t❛✐♥ ❡✈❡♥ ℓ✕❝②❝❧❡s ✇❤❡r❡ ℓ = 6, 8, . . . , n! ❤❛s ❤✐❡r❛r❝❤✐❝❛❧ str✉❝t✉r❡ ❤❛s ✐♥t❡❣r❛❧ s♣❡❝tr✉♠
❊❧❡♥❛ ❑♦♥st❛♥t✐♥♦✈❛ Pr♦♣❡rt✐❡s ♦❢ t❤❡ ❙t❛r ❣r❛♣❤s ❙❏❚❯✲✷✵✶✽ ✷ ✴ ✸✺
❚❤❡ ❙t❛r ❣r❛♣❤ Sn✱ n 3✱ ❝♦♥t❛✐♥s n ❝♦♣✐❡s Sn−1(i)✱ 1 i n✱ ✇❤❡r❡ ❡❛❝❤ Sn−1 ✐s ❛♥ ✐♥❞✉❝❡❞ s✉❜❣r❛♣❤ ♦♥ t❤❡ ✈❡rt❡① s❡t Vi = {[π1 . . . πn−1i], πk ∈ {1, . . . , n}\{i}, 1 k n − 1}✱ |Vi| = (n − 1)!
❊①❛♠♣❧❡✿ S4 ❤❛s ❢♦✉r ❝♦♣✐❡s ♦❢ S3
❊❧❡♥❛ ❑♦♥st❛♥t✐♥♦✈❛ Pr♦♣❡rt✐❡s ♦❢ t❤❡ ❙t❛r ❣r❛♣❤s ❙❏❚❯✲✷✵✶✽ ✸ ✴ ✸✺
❙♣❡❝tr✉♠ ♦❢ ❛ ❣r❛♣❤
❋♦r ❛ ❣r❛♣❤ Γ✱ t❤❡ s♣❡❝tr✉♠ ♦❢ Γ ✐s t❤❡ s♣❡❝tr✉♠ ♦❢ ✐ts ❛❞❥❛❝❡♥❝② ♠❛tr✐①✳
■♥t❡❣r❛❧ ❣r❛♣❤
❆ ❣r❛♣❤ ✐s ✐♥t❡❣r❛❧ ✐❢ ✐ts s♣❡❝tr✉♠ ❝♦♥s✐sts ❡♥t✐r❡❧② ♦❢ ✐♥t❡❣❡rs✳
❋✳ ❍❛r❛r② ❛♥❞ ❆✳ ❏✳ ❙❝❤✇❡♥❦✱ ❲❤✐❝❤ ❣r❛♣❤s ❤❛✈❡ ✐♥t❡❣r❛❧ s♣❡❝tr❛❄ ●r❛♣❤s ❛♥❞ ❈♦♠❜✐♥✳ ✭✶✾✼✹✮
■♥ ❋✳ ❍❛r❛r② ❛♥❞ ❆✳ ❏✳ ❙❝❤✇❡♥❦ ♣♦s❡❞ t❤❡ ♣r♦❜❧❡♠ ♦❢ ❝❤❛r❛❝t❡r✐③✐♥❣ t❤❡ ✐♥t❡❣r❛❧ ❣r❛♣❤s✳
❙♦♠❡ ❡①❛♠♣❧❡s ♦❢ ✐♥t❡❣r❛❧ ❣r❛♣❤s✿ ❝②❝❧❡s
✿ ✿ ✿
❊❧❡♥❛ ❑♦♥st❛♥t✐♥♦✈❛ Pr♦♣❡rt✐❡s ♦❢ t❤❡ ❙t❛r ❣r❛♣❤s ❙❏❚❯✲✷✵✶✽ ✹ ✴ ✸✺
❙♣❡❝tr✉♠ ♦❢ ❛ ❣r❛♣❤
❋♦r ❛ ❣r❛♣❤ Γ✱ t❤❡ s♣❡❝tr✉♠ ♦❢ Γ ✐s t❤❡ s♣❡❝tr✉♠ ♦❢ ✐ts ❛❞❥❛❝❡♥❝② ♠❛tr✐①✳
■♥t❡❣r❛❧ ❣r❛♣❤
❆ ❣r❛♣❤ Γ ✐s ✐♥t❡❣r❛❧ ✐❢ ✐ts s♣❡❝tr✉♠ ❝♦♥s✐sts ❡♥t✐r❡❧② ♦❢ ✐♥t❡❣❡rs✳
❋✳ ❍❛r❛r② ❛♥❞ ❆✳ ❏✳ ❙❝❤✇❡♥❦✱ ❲❤✐❝❤ ❣r❛♣❤s ❤❛✈❡ ✐♥t❡❣r❛❧ s♣❡❝tr❛❄ ●r❛♣❤s ❛♥❞ ❈♦♠❜✐♥✳ ✭✶✾✼✹✮
■♥ ❋✳ ❍❛r❛r② ❛♥❞ ❆✳ ❏✳ ❙❝❤✇❡♥❦ ♣♦s❡❞ t❤❡ ♣r♦❜❧❡♠ ♦❢ ❝❤❛r❛❝t❡r✐③✐♥❣ t❤❡ ✐♥t❡❣r❛❧ ❣r❛♣❤s✳
❙♦♠❡ ❡①❛♠♣❧❡s ♦❢ ✐♥t❡❣r❛❧ ❣r❛♣❤s✿ ❝②❝❧❡s
✿ ✿ ✿
❊❧❡♥❛ ❑♦♥st❛♥t✐♥♦✈❛ Pr♦♣❡rt✐❡s ♦❢ t❤❡ ❙t❛r ❣r❛♣❤s ❙❏❚❯✲✷✵✶✽ ✹ ✴ ✸✺
❙♣❡❝tr✉♠ ♦❢ ❛ ❣r❛♣❤
❋♦r ❛ ❣r❛♣❤ Γ✱ t❤❡ s♣❡❝tr✉♠ ♦❢ Γ ✐s t❤❡ s♣❡❝tr✉♠ ♦❢ ✐ts ❛❞❥❛❝❡♥❝② ♠❛tr✐①✳
■♥t❡❣r❛❧ ❣r❛♣❤
❆ ❣r❛♣❤ Γ ✐s ✐♥t❡❣r❛❧ ✐❢ ✐ts s♣❡❝tr✉♠ ❝♦♥s✐sts ❡♥t✐r❡❧② ♦❢ ✐♥t❡❣❡rs✳
❋✳ ❍❛r❛r② ❛♥❞ ❆✳ ❏✳ ❙❝❤✇❡♥❦✱ ❲❤✐❝❤ ❣r❛♣❤s ❤❛✈❡ ✐♥t❡❣r❛❧ s♣❡❝tr❛❄ ●r❛♣❤s ❛♥❞ ❈♦♠❜✐♥✳ ✭✶✾✼✹✮
■♥ 1974 ❋✳ ❍❛r❛r② ❛♥❞ ❆✳ ❏✳ ❙❝❤✇❡♥❦ ♣♦s❡❞ t❤❡ ♣r♦❜❧❡♠ ♦❢ ❝❤❛r❛❝t❡r✐③✐♥❣ t❤❡ ✐♥t❡❣r❛❧ ❣r❛♣❤s✳
❙♦♠❡ ❡①❛♠♣❧❡s ♦❢ ✐♥t❡❣r❛❧ ❣r❛♣❤s✿ ❝②❝❧❡s
✿ ✿ ✿
❊❧❡♥❛ ❑♦♥st❛♥t✐♥♦✈❛ Pr♦♣❡rt✐❡s ♦❢ t❤❡ ❙t❛r ❣r❛♣❤s ❙❏❚❯✲✷✵✶✽ ✹ ✴ ✸✺
❙♣❡❝tr✉♠ ♦❢ ❛ ❣r❛♣❤
❋♦r ❛ ❣r❛♣❤ Γ✱ t❤❡ s♣❡❝tr✉♠ ♦❢ Γ ✐s t❤❡ s♣❡❝tr✉♠ ♦❢ ✐ts ❛❞❥❛❝❡♥❝② ♠❛tr✐①✳
■♥t❡❣r❛❧ ❣r❛♣❤
❆ ❣r❛♣❤ Γ ✐s ✐♥t❡❣r❛❧ ✐❢ ✐ts s♣❡❝tr✉♠ ❝♦♥s✐sts ❡♥t✐r❡❧② ♦❢ ✐♥t❡❣❡rs✳
❋✳ ❍❛r❛r② ❛♥❞ ❆✳ ❏✳ ❙❝❤✇❡♥❦✱ ❲❤✐❝❤ ❣r❛♣❤s ❤❛✈❡ ✐♥t❡❣r❛❧ s♣❡❝tr❛❄ ●r❛♣❤s ❛♥❞ ❈♦♠❜✐♥✳ ✭✶✾✼✹✮
■♥ 1974 ❋✳ ❍❛r❛r② ❛♥❞ ❆✳ ❏✳ ❙❝❤✇❡♥❦ ♣♦s❡❞ t❤❡ ♣r♦❜❧❡♠ ♦❢ ❝❤❛r❛❝t❡r✐③✐♥❣ t❤❡ ✐♥t❡❣r❛❧ ❣r❛♣❤s✳
❙♦♠❡ ❡①❛♠♣❧❡s ♦❢ ✐♥t❡❣r❛❧ ❣r❛♣❤s✿ ❝②❝❧❡s
C3✿ (−12, 2) C4✿ (−2, 02, 2) C6✿ (−2, −12, 12, 2)
❊❧❡♥❛ ❑♦♥st❛♥t✐♥♦✈❛ Pr♦♣❡rt✐❡s ♦❢ t❤❡ ❙t❛r ❣r❛♣❤s ❙❏❚❯✲✷✵✶✽ ✹ ✴ ✸✺
❋✳❈✳❇✉ss❡♠❛❦❡r✱❉✳❈✈❡t❦♦✈✐✟ ❝ ✭✶✾✼✻✮❀❆✳❏✳❙❝❤✇❡♥❦ ✭✶✾✼✽✮ ❚❤❡r❡ ❛r❡ ❡①❛❝t❧② 13 ❝♦♥♥❡❝t❡❞✱ ❝✉❜✐❝✱ ✐♥t❡❣r❛❧ ❣r❛♣❤s
8 ❜✐♣❛rt✐t❡ ❝✉❜✐❝ ❣r❛♣❤s✿ p = 6 (±3, 04) p = 8 (±3, (±1)3) p = 10 (±3, ±2, (±1)2, 02) p = 12 (±3, (±2)2, ±1, 04) L6 p = 20 (±3, (±2)4, (±1)5) G9 ✭❇❈✬✼✻✮ p = 20 (±3, (±2)4, (±1)5) G10 ✭❇❈✬✼✻✮ p = 24 (±3, (±2)6, (±1)3, 04) S4 p = 30 (±3, (±2)9, 010)
❑♥♦✇♥ ❢❛❝ts
■❢ Γ ✐s ❛ ❜✐♣❛rt✐t❡ ❣r❛♣❤✱ ❛♥❞ λ ✐s ✐ts ❡✐❣❡♥✈❛❧✉❡ ✇✐t❤ ♠✉❧t✐♣❧✐❝✐t② mul(λ)✱ t❤❡♥ −λ ✐s ❛❧s♦ ✐ts ❡✐❣❡♥✈❛❧✉❡ ✇✐t❤ t❤❡ s❛♠❡ ♠✉❧t✐♣❧✐❝✐t②✳
❊❧❡♥❛ ❑♦♥st❛♥t✐♥♦✈❛ Pr♦♣❡rt✐❡s ♦❢ t❤❡ ❙t❛r ❣r❛♣❤s ❙❏❚❯✲✷✵✶✽ ✺ ✴ ✸✺
❑✳ ❇❛❧✐✁ ♥s❦❛✱ ❉✳ ❈✈❡t❦♦✈✐✁ ❝✱ ▼✳ ▲❡♣♦✈✐✁ ❝✱ ❙✳ ❙✐♠✐✁ ❝✱ ▼✳ ❑✉♣❝③②❦✱ ❑✳❚✳ ❩✇✐❡r③②✁ ♥s❦✐✱ ✭✶✾✾✾✲✷✵✵✵✮
❯s✐♥❣ ❇r❡♥❞❛♥ ▼❝❑❛②✬s ♣r♦❣r❛♠ GENG ❢♦r ❣❡♥❡r❛t✐♥❣ ❣r❛♣❤s✱ ✐t ✇❛s s❤♦✇♥ t❤❛t t❤❡r❡ ❛r❡ ❡①❛❝t❧② 263 ❝♦♥♥❡❝t❡❞ ✐♥t❡❣r❛❧ ❣r❛♣❤s ♦♥ ✉♣ t♦ 11 ✈❡rt✐❝❡s✳
❖✳ ❆❤♠❛❞✐✱ ◆✳ ❆❧♦♥✱ ■✳ ❋✳ ❇❧❛❦❡✱ ❛♥❞ ■✳ ❊✳ ❙❤♣❛r❧✐♥s❦✐✱
▼♦st ❣r❛♣❤s ❤❛✈❡ ♥♦♥✐♥t❡❣r❛❧ ❡✐❣❡♥✈❛❧✉❡s✱ ♠♦r❡ ♣r❡❝✐s❡❧②✱ ✐t ✇❛s ♣r♦✈❡❞ t❤❛t t❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ ❛ ❧❛❜❡❧❡❞ ❣r❛♣❤ ♦♥ ✈❡rt✐❝❡s t♦ ❜❡ ✐♥t❡❣r❛❧ ✐s ❛t ♠♦st ❢♦r ❛ s✉✣❝✐❡♥t❧② ❧❛r❣❡ ✳
❖✳ ❆❤♠❛❞✐✱ ◆✳ ❆❧♦♥✱ ■✳ ❋✳ ❇❧❛❦❡✱■✳ ❊✳ ❙❤♣❛r❧✐♥s❦✐
❲❡ ❜❡❧✐❡✈❡ ♦✉r ❜♦✉♥❞ ✐s ❢❛r ❢r♦♠ ❜❡✐♥❣ t✐❣❤t ❛♥❞ t❤❡ ♥✉♠❜❡r ♦❢ ✐♥t❡❣r❛❧ ❣r❛♣❤s ✐s s✉❜st❛♥t✐❛❧❧② s♠❛❧❧❡r✳
❊❧❡♥❛ ❑♦♥st❛♥t✐♥♦✈❛ Pr♦♣❡rt✐❡s ♦❢ t❤❡ ❙t❛r ❣r❛♣❤s ❙❏❚❯✲✷✵✶✽ ✻ ✴ ✸✺
❑✳ ❇❛❧✐✁ ♥s❦❛✱ ❉✳ ❈✈❡t❦♦✈✐✁ ❝✱ ▼✳ ▲❡♣♦✈✐✁ ❝✱ ❙✳ ❙✐♠✐✁ ❝✱ ▼✳ ❑✉♣❝③②❦✱ ❑✳❚✳ ❩✇✐❡r③②✁ ♥s❦✐✱ ✭✶✾✾✾✲✷✵✵✵✮
❯s✐♥❣ ❇r❡♥❞❛♥ ▼❝❑❛②✬s ♣r♦❣r❛♠ GENG ❢♦r ❣❡♥❡r❛t✐♥❣ ❣r❛♣❤s✱ ✐t ✇❛s s❤♦✇♥ t❤❛t t❤❡r❡ ❛r❡ ❡①❛❝t❧② 263 ❝♦♥♥❡❝t❡❞ ✐♥t❡❣r❛❧ ❣r❛♣❤s ♦♥ ✉♣ t♦ 11 ✈❡rt✐❝❡s✳
❖✳ ❆❤♠❛❞✐✱ ◆✳ ❆❧♦♥✱ ■✳ ❋✳ ❇❧❛❦❡✱ ❛♥❞ ■✳ ❊✳ ❙❤♣❛r❧✐♥s❦✐✱
▼♦st ❣r❛♣❤s ❤❛✈❡ ♥♦♥✐♥t❡❣r❛❧ ❡✐❣❡♥✈❛❧✉❡s✱ ♠♦r❡ ♣r❡❝✐s❡❧②✱ ✐t ✇❛s ♣r♦✈❡❞ t❤❛t t❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ ❛ ❧❛❜❡❧❡❞ ❣r❛♣❤ ♦♥ n ✈❡rt✐❝❡s t♦ ❜❡ ✐♥t❡❣r❛❧ ✐s ❛t ♠♦st 2−n/400 ❢♦r ❛ s✉✣❝✐❡♥t❧② ❧❛r❣❡ n✳
❖✳ ❆❤♠❛❞✐✱ ◆✳ ❆❧♦♥✱ ■✳ ❋✳ ❇❧❛❦❡✱■✳ ❊✳ ❙❤♣❛r❧✐♥s❦✐
❲❡ ❜❡❧✐❡✈❡ ♦✉r ❜♦✉♥❞ ✐s ❢❛r ❢r♦♠ ❜❡✐♥❣ t✐❣❤t ❛♥❞ t❤❡ ♥✉♠❜❡r ♦❢ ✐♥t❡❣r❛❧ ❣r❛♣❤s ✐s s✉❜st❛♥t✐❛❧❧② s♠❛❧❧❡r✳
❊❧❡♥❛ ❑♦♥st❛♥t✐♥♦✈❛ Pr♦♣❡rt✐❡s ♦❢ t❤❡ ❙t❛r ❣r❛♣❤s ❙❏❚❯✲✷✵✶✽ ✻ ✴ ✸✺
❑✳ ❇❛❧✐✁ ♥s❦❛✱ ❉✳ ❈✈❡t❦♦✈✐✁ ❝✱ ▼✳ ▲❡♣♦✈✐✁ ❝✱ ❙✳ ❙✐♠✐✁ ❝✱ ▼✳ ❑✉♣❝③②❦✱ ❑✳❚✳ ❩✇✐❡r③②✁ ♥s❦✐✱ ✭✶✾✾✾✲✷✵✵✵✮
❯s✐♥❣ ❇r❡♥❞❛♥ ▼❝❑❛②✬s ♣r♦❣r❛♠ GENG ❢♦r ❣❡♥❡r❛t✐♥❣ ❣r❛♣❤s✱ ✐t ✇❛s s❤♦✇♥ t❤❛t t❤❡r❡ ❛r❡ ❡①❛❝t❧② 263 ❝♦♥♥❡❝t❡❞ ✐♥t❡❣r❛❧ ❣r❛♣❤s ♦♥ ✉♣ t♦ 11 ✈❡rt✐❝❡s✳
❖✳ ❆❤♠❛❞✐✱ ◆✳ ❆❧♦♥✱ ■✳ ❋✳ ❇❧❛❦❡✱ ❛♥❞ ■✳ ❊✳ ❙❤♣❛r❧✐♥s❦✐✱
▼♦st ❣r❛♣❤s ❤❛✈❡ ♥♦♥✐♥t❡❣r❛❧ ❡✐❣❡♥✈❛❧✉❡s✱ ♠♦r❡ ♣r❡❝✐s❡❧②✱ ✐t ✇❛s ♣r♦✈❡❞ t❤❛t t❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ ❛ ❧❛❜❡❧❡❞ ❣r❛♣❤ ♦♥ n ✈❡rt✐❝❡s t♦ ❜❡ ✐♥t❡❣r❛❧ ✐s ❛t ♠♦st 2−n/400 ❢♦r ❛ s✉✣❝✐❡♥t❧② ❧❛r❣❡ n✳
❖✳ ❆❤♠❛❞✐✱ ◆✳ ❆❧♦♥✱ ■✳ ❋✳ ❇❧❛❦❡✱■✳ ❊✳ ❙❤♣❛r❧✐♥s❦✐
❲❡ ❜❡❧✐❡✈❡ ♦✉r ❜♦✉♥❞ ✐s ❢❛r ❢r♦♠ ❜❡✐♥❣ t✐❣❤t ❛♥❞ t❤❡ ♥✉♠❜❡r ♦❢ ✐♥t❡❣r❛❧ ❣r❛♣❤s ✐s s✉❜st❛♥t✐❛❧❧② s♠❛❧❧❡r✳
❊❧❡♥❛ ❑♦♥st❛♥t✐♥♦✈❛ Pr♦♣❡rt✐❡s ♦❢ t❤❡ ❙t❛r ❣r❛♣❤s ❙❏❚❯✲✷✵✶✽ ✻ ✴ ✸✺
❆✳ ❆❜❞♦❧❧❛❤✐✱ ❊✳ ❱❛t❛♥❞♦♦st ✭✷✵✵✾✮ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ✐♥t❡❣r❛❧ ❝✉❜✐❝ ❈❛②❧❡② ❣r❛♣❤s
❚❤❡r❡ ❛r❡ ❡①❛❝t❧② s❡✈❡♥ ❝♦♥♥❡❝t❡❞ ❝✉❜✐❝ ✐♥t❡❣r❛❧ ❈❛②❧❡② ❣r❛♣❤s✳ ■♥ ♣❛rt✐❝✉❧❛r✱ ❢♦r ❛ ✜♥✐t❡ ❣r♦✉♣ G ❛♥❞ ❛ ❣❡♥❡r❛t✐♥❣ s❡t S, |S| = 3, t❤❡ ❈❛②❧❡② ❣r❛♣❤ Γ ✐s ✐♥t❡❣r❛❧ ✐❢ ❛♥❞ ♦♥❧② ✐❢ G ✐s ✐s♦♠♦r♣❤✐❝ t♦ ♦♥❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❣r♦✉♣s✿ C 2
2 ✱ C4✱ C6✱ Sym3✱ C 3 2 ✱ C2 × C4✱ D8✱ C2 × C6✱ D12✱ A4✱ Sym4✱
D8 × C3✱ D6 × C4 ♦r A4 × C2✳ ✐s t❤❡ ❝②❝❧✐❝ ❣r♦✉♣ ♦❢ ♦r❞❡r ✐s t❤❡ ❞✐❤❡❞r❛❧ ❣r♦✉♣ ♦❢ ♦r❞❡r ✐s t❤❡ s②♠♠❡tr✐❝ ❣r♦✉♣ ♦❢ ♦r❞❡r ✐s t❤❡ ❛❧t❡r♥❛t✐♥❣ ❣r♦✉♣ ♦❢ ♦r❞❡r
❊❧❡♥❛ ❑♦♥st❛♥t✐♥♦✈❛ Pr♦♣❡rt✐❡s ♦❢ t❤❡ ❙t❛r ❣r❛♣❤s ❙❏❚❯✲✷✵✶✽ ✼ ✴ ✸✺
❆✳ ❆❜❞♦❧❧❛❤✐✱ ❊✳ ❱❛t❛♥❞♦♦st ✭✷✵✵✾✮ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ✐♥t❡❣r❛❧ ❝✉❜✐❝ ❈❛②❧❡② ❣r❛♣❤s
❚❤❡r❡ ❛r❡ ❡①❛❝t❧② s❡✈❡♥ ❝♦♥♥❡❝t❡❞ ❝✉❜✐❝ ✐♥t❡❣r❛❧ ❈❛②❧❡② ❣r❛♣❤s✳ ■♥ ♣❛rt✐❝✉❧❛r✱ ❢♦r ❛ ✜♥✐t❡ ❣r♦✉♣ G ❛♥❞ ❛ ❣❡♥❡r❛t✐♥❣ s❡t S, |S| = 3, t❤❡ ❈❛②❧❡② ❣r❛♣❤ Γ ✐s ✐♥t❡❣r❛❧ ✐❢ ❛♥❞ ♦♥❧② ✐❢ G ✐s ✐s♦♠♦r♣❤✐❝ t♦ ♦♥❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❣r♦✉♣s✿ C 2
2 ✱ C4✱ C6✱ Sym3✱ C 3 2 ✱ C2 × C4✱ D8✱ C2 × C6✱ D12✱ A4✱ Sym4✱
D8 × C3✱ D6 × C4 ♦r A4 × C2✳ Cn ✐s t❤❡ ❝②❝❧✐❝ ❣r♦✉♣ ♦❢ ♦r❞❡r n D2n ✐s t❤❡ ❞✐❤❡❞r❛❧ ❣r♦✉♣ ♦❢ ♦r❞❡r 2n, n > 2 Symn ✐s t❤❡ s②♠♠❡tr✐❝ ❣r♦✉♣ ♦❢ ♦r❞❡r n An ✐s t❤❡ ❛❧t❡r♥❛t✐♥❣ ❣r♦✉♣ ♦❢ ♦r❞❡r n
❊❧❡♥❛ ❑♦♥st❛♥t✐♥♦✈❛ Pr♦♣❡rt✐❡s ♦❢ t❤❡ ❙t❛r ❣r❛♣❤s ❙❏❚❯✲✷✵✶✽ ✼ ✴ ✸✺
❊❧❡♥❛ ❑♦♥st❛♥t✐♥♦✈❛ Pr♦♣❡rt✐❡s ♦❢ t❤❡ ❙t❛r ❣r❛♣❤s ❙❏❚❯✲✷✵✶✽ ✽ ✴ ✸✺
❈❤❛r❛❝t❡r✐③❛t✐♦♥ ♦❢ ✐♥t❡❣r❛❧ ❈❛②❧❡② ❣r❛♣❤s
❍❛♠♠✐♥❣ ❣r❛♣❤s H(n, q)✿ λm = n(q − 1) − qm✱ ✇❤❡r❡ m = 0, 1, ..., n✱ ✇✐t❤ ♠✉❧t✐♣❧✐❝✐t✐❡s n
m
❈❛②❧❡② ❣r❛♣❤s ♦✈❡r ❝②❝❧✐❝ ❣r♦✉♣s ✭❝✐r❝✉❧❛♥ts✮ ✭❲✳ ❙♦✱ 2005✮ ❈❛②❧❡② ❣r❛♣❤s ♦✈❡r ❛❜❡❧✐❛♥ ❣r♦✉♣s ✭❲✳ ❑❧♦t③✱ ❚✳ ❙❛♥❞❡r✱ 2010✮ ❈❛②❧❡② ❣r❛♣❤s ♦✈❡r ❞✐❤❡❞r❛❧ ❣r♦✉♣s ✭▲✉ ▲✉✱ ◗✐♦♥❣①✐❛♥❣ ❍✉❛♥❣✱ ❳✉❡②✐ ❍✉❛♥❣✱ 2017✮
❍❛♠♠✐♥❣ ❣r❛♣❤s ❛r❡ ❈❛②❧❡② ❣r❛♣❤s
■♥ ♣❛rt✐❝✉❧❛r✱ t❤❡ ❤②♣❡r❝✉❜❡ ❣r❛♣❤s Hn ✭❜✐♥❛r② ❝❛s❡ ♦❢ t❤❡ ❍❛♠♠✐♥❣ ❣r❛♣❤s✮ ❛r❡ ❈❛②❧❡② ❣r❛♣❤ ♦♥ t❤❡ ❣r♦✉♣ Zn
2 ✇✐t❤ t❤❡ ❣❡♥❡r❛t✐♥❣ s❡t
S = {(0, . . . , 0
i
, 1, 0, . . . , 0
n−i−1
), 0 i n − 1}✳
❊❧❡♥❛ ❑♦♥st❛♥t✐♥♦✈❛ Pr♦♣❡rt✐❡s ♦❢ t❤❡ ❙t❛r ❣r❛♣❤s ❙❏❚❯✲✷✵✶✽ ✾ ✴ ✸✺
❈♦♥❥❡❝t✉r❡ ✭❆✳ ❆❜❞♦❧❧❛❤✐ ❛♥❞ ❊✳ ❱❛t❛♥❞♦♦st✱ ✷✵✵✾✮
❚❤❡ s♣❡❝tr✉♠ ♦❢ Sn ✐s ✐♥t❡❣r❛❧✱ ❛♥❞ ❝♦♥t❛✐♥s ❛❧❧ ✐♥t❡❣❡rs ✐♥ t❤❡ r❛♥❣❡ ❢r♦♠ −(n − 1) ✉♣ t♦ n − 1 ✭✇✐t❤ t❤❡ s♦❧❡ ❡①❝❡♣t✐♦♥ t❤❛t ✇❤❡♥ n 3✱ ③❡r♦ ✐s ♥♦t ❛♥ ❡✐❣❡♥✈❛❧✉❡ ♦❢ Sn✮✳
P❛rt✐❛❧❧② t❤✐s ❝♦♥❥❡❝t✉r❡ ✇❛s ❜❛s❡❞ ♦♥ t❤❡ ❦♥♦✇♥ ❢❛❝ts
❋❛❝t ✶✳ ■❢ ✐s ❛ ✲r❡❣✉❧❛r ❣r❛♣❤✱ ❛♥❞ ✐s ❛♥ ❡✐❣❡♥✈❛❧✉❡ ♦❢ ✐ts ❛❞❥❛❝❡♥❝② ♠❛tr✐①✱ t❤❡♥ ✳ ❋❛❝t ✷✳ ■❢ ✐s ❛ ❜✐♣❛rt✐t❡ ❣r❛♣❤✱ ❛♥❞ ✐s ✐ts ❡✐❣❡♥✈❛❧✉❡ ✇✐t❤ ♠✉❧t✐♣❧✐❝✐t② ✱ t❤❡♥ ✐s ❛❧s♦ ✐ts ❡✐❣❡♥✈❛❧✉❡ ✇✐t❤ t❤❡ s❛♠❡ ♠✉❧t✐♣❧✐❝✐t②✳ ❋♦r ✱ t❤❡ ❝♦♥❥❡❝t✉r❡ ✇❛s ✈❡r✐✜❡❞ ❜② ●❆P
❊❧❡♥❛ ❑♦♥st❛♥t✐♥♦✈❛ Pr♦♣❡rt✐❡s ♦❢ t❤❡ ❙t❛r ❣r❛♣❤s ❙❏❚❯✲✷✵✶✽ ✶✵ ✴ ✸✺
❈♦♥❥❡❝t✉r❡ ✭❆✳ ❆❜❞♦❧❧❛❤✐ ❛♥❞ ❊✳ ❱❛t❛♥❞♦♦st✱ ✷✵✵✾✮
❚❤❡ s♣❡❝tr✉♠ ♦❢ Sn ✐s ✐♥t❡❣r❛❧✱ ❛♥❞ ❝♦♥t❛✐♥s ❛❧❧ ✐♥t❡❣❡rs ✐♥ t❤❡ r❛♥❣❡ ❢r♦♠ −(n − 1) ✉♣ t♦ n − 1 ✭✇✐t❤ t❤❡ s♦❧❡ ❡①❝❡♣t✐♦♥ t❤❛t ✇❤❡♥ n 3✱ ③❡r♦ ✐s ♥♦t ❛♥ ❡✐❣❡♥✈❛❧✉❡ ♦❢ Sn✮✳
P❛rt✐❛❧❧② t❤✐s ❝♦♥❥❡❝t✉r❡ ✇❛s ❜❛s❡❞ ♦♥ t❤❡ ❦♥♦✇♥ ❢❛❝ts
❋❛❝t ✶✳ ■❢ Γ ✐s ❛ r✲r❡❣✉❧❛r ❣r❛♣❤✱ ❛♥❞ λ ✐s ❛♥ ❡✐❣❡♥✈❛❧✉❡ ♦❢ ✐ts ❛❞❥❛❝❡♥❝② ♠❛tr✐①✱ t❤❡♥ |λ| r✳ ❋❛❝t ✷✳ ■❢ Γ ✐s ❛ ❜✐♣❛rt✐t❡ ❣r❛♣❤✱ ❛♥❞ λ ✐s ✐ts ❡✐❣❡♥✈❛❧✉❡ ✇✐t❤ ♠✉❧t✐♣❧✐❝✐t② mul(λ)✱ t❤❡♥ −λ ✐s ❛❧s♦ ✐ts ❡✐❣❡♥✈❛❧✉❡ ✇✐t❤ t❤❡ s❛♠❡ ♠✉❧t✐♣❧✐❝✐t②✳ ❋♦r ✱ t❤❡ ❝♦♥❥❡❝t✉r❡ ✇❛s ✈❡r✐✜❡❞ ❜② ●❆P
❊❧❡♥❛ ❑♦♥st❛♥t✐♥♦✈❛ Pr♦♣❡rt✐❡s ♦❢ t❤❡ ❙t❛r ❣r❛♣❤s ❙❏❚❯✲✷✵✶✽ ✶✵ ✴ ✸✺
❈♦♥❥❡❝t✉r❡ ✭❆✳ ❆❜❞♦❧❧❛❤✐ ❛♥❞ ❊✳ ❱❛t❛♥❞♦♦st✱ ✷✵✵✾✮
❚❤❡ s♣❡❝tr✉♠ ♦❢ Sn ✐s ✐♥t❡❣r❛❧✱ ❛♥❞ ❝♦♥t❛✐♥s ❛❧❧ ✐♥t❡❣❡rs ✐♥ t❤❡ r❛♥❣❡ ❢r♦♠ −(n − 1) ✉♣ t♦ n − 1 ✭✇✐t❤ t❤❡ s♦❧❡ ❡①❝❡♣t✐♦♥ t❤❛t ✇❤❡♥ n 3✱ ③❡r♦ ✐s ♥♦t ❛♥ ❡✐❣❡♥✈❛❧✉❡ ♦❢ Sn✮✳
P❛rt✐❛❧❧② t❤✐s ❝♦♥❥❡❝t✉r❡ ✇❛s ❜❛s❡❞ ♦♥ t❤❡ ❦♥♦✇♥ ❢❛❝ts
❋❛❝t ✶✳ ■❢ Γ ✐s ❛ r✲r❡❣✉❧❛r ❣r❛♣❤✱ ❛♥❞ λ ✐s ❛♥ ❡✐❣❡♥✈❛❧✉❡ ♦❢ ✐ts ❛❞❥❛❝❡♥❝② ♠❛tr✐①✱ t❤❡♥ |λ| r✳ ❋❛❝t ✷✳ ■❢ Γ ✐s ❛ ❜✐♣❛rt✐t❡ ❣r❛♣❤✱ ❛♥❞ λ ✐s ✐ts ❡✐❣❡♥✈❛❧✉❡ ✇✐t❤ ♠✉❧t✐♣❧✐❝✐t② mul(λ)✱ t❤❡♥ −λ ✐s ❛❧s♦ ✐ts ❡✐❣❡♥✈❛❧✉❡ ✇✐t❤ t❤❡ s❛♠❡ ♠✉❧t✐♣❧✐❝✐t②✳ ❋♦r n 6✱ t❤❡ ❝♦♥❥❡❝t✉r❡ ✇❛s ✈❡r✐✜❡❞ ❜② ●❆P
❊❧❡♥❛ ❑♦♥st❛♥t✐♥♦✈❛ Pr♦♣❡rt✐❡s ♦❢ t❤❡ ❙t❛r ❣r❛♣❤s ❙❏❚❯✲✷✵✶✽ ✶✵ ✴ ✸✺
❘✳ ❑r❛❦♦✈s❦✐ ❛♥❞ ❇✳ ▼♦❤❛r ♣r♦✈❡❞ t❤❡ s❡❝♦♥❞ ♣❛rt ♦❢ t❤❡ ❝♦♥❥❡❝t✉r❡✳
❚❤❡♦r❡♠ ✭❘✳ ❑r❛❦♦✈s❦✐ ❛♥❞ ❇✳ ▼♦❤❛r✱ ✷✵✶✷✮
▲❡t ✱ t❤❡♥ ❢♦r ❡❛❝❤ ✐♥t❡❣❡r t❤❡ ✈❛❧✉❡s ❛r❡ ❡✐❣❡♥✈❛❧✉❡s ♦❢ ✇✐t❤ ♠✉❧t✐♣❧✐❝✐t② ❛t ❧❡❛st ✳ ■❢ ✱ t❤❡♥ ✐s ❛♥ ❡✐❣❡♥✈❛❧✉❡ ♦❢ ✇✐t❤ ♠✉❧t✐♣❧✐❝✐t② ❛t ❧❡❛st ✳ ✵ ✷ ✶ ✸ ✷ ✶ ✹ ✹ ✸ ✻ ✶ ✺ ✸✵ ✹ ✷✽ ✶✷ ✶ ✻ ✶✻✽ ✸✵ ✶✷✵ ✶✵✺ ✷✵ ✶
❊❧❡♥❛ ❑♦♥st❛♥t✐♥♦✈❛ Pr♦♣❡rt✐❡s ♦❢ t❤❡ ❙t❛r ❣r❛♣❤s ❙❏❚❯✲✷✵✶✽ ✶✶ ✴ ✸✺
❘✳ ❑r❛❦♦✈s❦✐ ❛♥❞ ❇✳ ▼♦❤❛r ♣r♦✈❡❞ t❤❡ s❡❝♦♥❞ ♣❛rt ♦❢ t❤❡ ❝♦♥❥❡❝t✉r❡✳
❚❤❡♦r❡♠ ✭❘✳ ❑r❛❦♦✈s❦✐ ❛♥❞ ❇✳ ▼♦❤❛r✱ ✷✵✶✷✮
▲❡t n 2✱ t❤❡♥ ❢♦r ❡❛❝❤ ✐♥t❡❣❡r 1 k n − 1 t❤❡ ✈❛❧✉❡s ±(n − k) ❛r❡ ❡✐❣❡♥✈❛❧✉❡s ♦❢ Sn ✇✐t❤ ♠✉❧t✐♣❧✐❝✐t② ❛t ❧❡❛st
k−1
❡✐❣❡♥✈❛❧✉❡ ♦❢ Sn ✇✐t❤ ♠✉❧t✐♣❧✐❝✐t② ❛t ❧❡❛st
2
✵ ✷ ✶ ✸ ✷ ✶ ✹ ✹ ✸ ✻ ✶ ✺ ✸✵ ✹ ✷✽ ✶✷ ✶ ✻ ✶✻✽ ✸✵ ✶✷✵ ✶✵✺ ✷✵ ✶
❊❧❡♥❛ ❑♦♥st❛♥t✐♥♦✈❛ Pr♦♣❡rt✐❡s ♦❢ t❤❡ ❙t❛r ❣r❛♣❤s ❙❏❚❯✲✷✵✶✽ ✶✶ ✴ ✸✺
❘✳ ❑r❛❦♦✈s❦✐ ❛♥❞ ❇✳ ▼♦❤❛r ♣r♦✈❡❞ t❤❡ s❡❝♦♥❞ ♣❛rt ♦❢ t❤❡ ❝♦♥❥❡❝t✉r❡✳
❚❤❡♦r❡♠ ✭❘✳ ❑r❛❦♦✈s❦✐ ❛♥❞ ❇✳ ▼♦❤❛r✱ ✷✵✶✷✮
▲❡t n 2✱ t❤❡♥ ❢♦r ❡❛❝❤ ✐♥t❡❣❡r 1 k n − 1 t❤❡ ✈❛❧✉❡s ±(n − k) ❛r❡ ❡✐❣❡♥✈❛❧✉❡s ♦❢ Sn ✇✐t❤ ♠✉❧t✐♣❧✐❝✐t② ❛t ❧❡❛st
k−1
❡✐❣❡♥✈❛❧✉❡ ♦❢ Sn ✇✐t❤ ♠✉❧t✐♣❧✐❝✐t② ❛t ❧❡❛st
2
n/λ
✵ ±1 ±2 ±3 ±4 ±5 ✷ ✶ ✸ ✷ ✶ ✹ ✹ ✸ ✻ ✶ ✺ ✸✵ ✹ ✷✽ ✶✷ ✶ ✻ ✶✻✽ ✸✵ ✶✷✵ ✶✵✺ ✷✵ ✶
❊❧❡♥❛ ❑♦♥st❛♥t✐♥♦✈❛ Pr♦♣❡rt✐❡s ♦❢ t❤❡ ❙t❛r ❣r❛♣❤s ❙❏❚❯✲✷✵✶✽ ✶✶ ✴ ✸✺
✇❛s ❛❧r❡❛❞② s♦❧✈❡❞ ✐♥ ❛♥♦t❤❡r ❝♦♥t❡①t✳
❚❤❡♦r❡♠ ✭●✳ ❈❤❛♣✉② ❛♥❞ ❱✳ ❋❡r❛②✱ ✷✵✶✷✮
❚❤❡ s♣❡❝tr✉♠ ♦❢ ❝♦♥t❛✐♥s ♦♥❧② ✐♥t❡❣❡rs✳ ❚❤❡ ♠✉❧t✐♣❧✐❝✐t② ✱ ✇❤❡r❡ ✱ ♦❢ ❛♥ ✐♥t❡❣❡r ✐s ❣✐✈❡♥ ❜②✿ ✇❤❡r❡ ✐s t❤❡ ❞✐♠❡♥s✐♦♥ ♦❢ ❛♥ ✐rr❡❞✉❝✐❜❧❡ ♠♦❞✉❧❡✱ ✐s t❤❡ ♥✉♠❜❡r ♦❢ st❛♥❞❛r❞ ❨♦✉♥❣ t❛❜❧❡❛✉① ♦❢ s❤❛♣❡ ✱ s❛t✐s❢②✐♥❣ ✳
❊❧❡♥❛ ❑♦♥st❛♥t✐♥♦✈❛ Pr♦♣❡rt✐❡s ♦❢ t❤❡ ❙t❛r ❣r❛♣❤s ❙❏❚❯✲✷✵✶✽ ✶✷ ✴ ✸✺
✇❛s ❛❧r❡❛❞② s♦❧✈❡❞ ✐♥ ❛♥♦t❤❡r ❝♦♥t❡①t✳
❚❤❡♦r❡♠ ✭●✳ ❈❤❛♣✉② ❛♥❞ ❱✳ ❋❡r❛②✱ ✷✵✶✷✮
❚❤❡ s♣❡❝tr✉♠ ♦❢ Sn ❝♦♥t❛✐♥s ♦♥❧② ✐♥t❡❣❡rs✳ ❚❤❡ ♠✉❧t✐♣❧✐❝✐t② mul(n − k)✱ ✇❤❡r❡ 1 k n − 1✱ ♦❢ ❛♥ ✐♥t❡❣❡r (n − k) ∈ Z ✐s ❣✐✈❡♥ ❜②✿ mul(n − k) =
dim(Vλ)Iλ(n − k), ✇❤❡r❡ dim(Vλ) ✐s t❤❡ ❞✐♠❡♥s✐♦♥ ♦❢ ❛♥ ✐rr❡❞✉❝✐❜❧❡ ♠♦❞✉❧❡✱ Iλ(n − k) ✐s t❤❡ ♥✉♠❜❡r ♦❢ st❛♥❞❛r❞ ❨♦✉♥❣ t❛❜❧❡❛✉① ♦❢ s❤❛♣❡ λ✱ s❛t✐s❢②✐♥❣ c(n) = n − k✳
❊❧❡♥❛ ❑♦♥st❛♥t✐♥♦✈❛ Pr♦♣❡rt✐❡s ♦❢ t❤❡ ❙t❛r ❣r❛♣❤s ❙❏❚❯✲✷✵✶✽ ✶✷ ✴ ✸✺
❈♦r♦❧❧❛r② ✭●✳ ❈❤❛♣✉② ❛♥❞ ❱✳ ❋❡r❛②✱ ✷✵✶✷✮
▲❡t n 2✱ t❤❡♥ ❢♦r ❡❛❝❤ ✐♥t❡❣❡r 1 k n − 1 t❤❡ ✈❛❧✉❡s ±(n − k) ❛r❡ ❡✐❣❡♥✈❛❧✉❡s ♦❢ Sn ✇✐t❤ ♠✉❧t✐♣❧✐❝✐t② ❛t ❧❡❛st
k−1 n−1 k
❚❤❡ ❜♦✉♥❞ ✐s ❛❝❤✐❡✈❡❞ ❢♦r ✳
❚❤❡♦r❡♠ ✭❊✳ ❑❤♦♠②❛❦♦✈❛✱ ❊❑✱ ✷✵✶✻✮
❚❤❡ ♠✉❧t✐♣❧✐❝✐t✐❡s ✱ ✇❤❡r❡ ❛♥❞ ✱ ♦❢ t❤❡ ❡✐❣❡♥✈❛❧✉❡s ♦❢ t❤❡ ❙t❛r ❣r❛♣❤ ❛r❡ ❣✐✈❡♥ ❜② t❤❡ ❢♦❧❧♦✇✐♥❣ ❢♦r♠✉❧❛s✿
❊❧❡♥❛ ❑♦♥st❛♥t✐♥♦✈❛ Pr♦♣❡rt✐❡s ♦❢ t❤❡ ❙t❛r ❣r❛♣❤s ❙❏❚❯✲✷✵✶✽ ✶✸ ✴ ✸✺
❈♦r♦❧❧❛r② ✭●✳ ❈❤❛♣✉② ❛♥❞ ❱✳ ❋❡r❛②✱ ✷✵✶✷✮
▲❡t n 2✱ t❤❡♥ ❢♦r ❡❛❝❤ ✐♥t❡❣❡r 1 k n − 1 t❤❡ ✈❛❧✉❡s ±(n − k) ❛r❡ ❡✐❣❡♥✈❛❧✉❡s ♦❢ Sn ✇✐t❤ ♠✉❧t✐♣❧✐❝✐t② ❛t ❧❡❛st
k−1 n−1 k
❚❤❡ ❜♦✉♥❞ ✐s ❛❝❤✐❡✈❡❞ ❢♦r k = 2✳
❚❤❡♦r❡♠ ✭❊✳ ❑❤♦♠②❛❦♦✈❛✱ ❊❑✱ ✷✵✶✻✮
❚❤❡ ♠✉❧t✐♣❧✐❝✐t✐❡s ✱ ✇❤❡r❡ ❛♥❞ ✱ ♦❢ t❤❡ ❡✐❣❡♥✈❛❧✉❡s ♦❢ t❤❡ ❙t❛r ❣r❛♣❤ ❛r❡ ❣✐✈❡♥ ❜② t❤❡ ❢♦❧❧♦✇✐♥❣ ❢♦r♠✉❧❛s✿
❊❧❡♥❛ ❑♦♥st❛♥t✐♥♦✈❛ Pr♦♣❡rt✐❡s ♦❢ t❤❡ ❙t❛r ❣r❛♣❤s ❙❏❚❯✲✷✵✶✽ ✶✸ ✴ ✸✺
❈♦r♦❧❧❛r② ✭●✳ ❈❤❛♣✉② ❛♥❞ ❱✳ ❋❡r❛②✱ ✷✵✶✷✮
▲❡t n 2✱ t❤❡♥ ❢♦r ❡❛❝❤ ✐♥t❡❣❡r 1 k n − 1 t❤❡ ✈❛❧✉❡s ±(n − k) ❛r❡ ❡✐❣❡♥✈❛❧✉❡s ♦❢ Sn ✇✐t❤ ♠✉❧t✐♣❧✐❝✐t② ❛t ❧❡❛st
k−1 n−1 k
❚❤❡ ❜♦✉♥❞ ✐s ❛❝❤✐❡✈❡❞ ❢♦r k = 2✳
❚❤❡♦r❡♠ ✭❊✳ ❑❤♦♠②❛❦♦✈❛✱ ❊❑✱ ✷✵✶✻✮
❚❤❡ ♠✉❧t✐♣❧✐❝✐t✐❡s mul(n − k)✱ ✇❤❡r❡ k = 2, 3, 4, 5 ❛♥❞ n 2k − 1✱ ♦❢ t❤❡ ❡✐❣❡♥✈❛❧✉❡s (n − k) ♦❢ t❤❡ ❙t❛r ❣r❛♣❤ Sn ❛r❡ ❣✐✈❡♥ ❜② t❤❡ ❢♦❧❧♦✇✐♥❣ ❢♦r♠✉❧❛s✿ mul(n − 2) = (n − 1)(n − 2) mul(n − 3) = (n−3)(n−1)
2
(n2 − 4n + 2) mul(n − 4) = (n−2)(n−1)
6
(n4 − 12n3 + 47n2 − 62n + 12) mul(n − 5) = (n−2)(n−1)
24
(n6 − 21n5 + 169n4 − 647n3 + 1174n2 − 820n + 60)
❊❧❡♥❛ ❑♦♥st❛♥t✐♥♦✈❛ Pr♦♣❡rt✐❡s ♦❢ t❤❡ ❙t❛r ❣r❛♣❤s ❙❏❚❯✲✷✵✶✽ ✶✸ ✴ ✸✺
❉❡✜♥✐t✐♦♥
▲❡t G = (V , E) ❜❡ ❛ s✐♠♣❧❡ ❣r❛♣❤✳ ❆ ❢✉♥❝t✐♦♥ f : V − → R ✐s ❝❛❧❧❡❞ ❛♥ ❡✐❣❡♥❢✉♥❝t✐♦♥ ♦❢ t❤❡ ❣r❛♣❤ G ❝♦rr❡s♣♦♥❞✐♥❣ t♦ ❛♥ ❡✐❣❡♥✈❛❧✉❡ λ ✐❢ f ≡ 0 ❛♥❞ ❢♦r ❛♥② ✈❡rt❡① x t❤❡ ❧♦❝❛❧ ❝♦♥❞✐t✐♦♥ λ · f (x) =
y∈N(x) f (y) ❤♦❧❞s✳
✐s ♥❡✐❣❤❜♦r❤♦♦❞
❊❧❡♥❛ ❑♦♥st❛♥t✐♥♦✈❛ Pr♦♣❡rt✐❡s ♦❢ t❤❡ ❙t❛r ❣r❛♣❤s ❙❏❚❯✲✷✵✶✽ ✶✹ ✴ ✸✺
❉❡✜♥✐t✐♦♥
▲❡t G = (V , E) ❜❡ ❛ s✐♠♣❧❡ ❣r❛♣❤✳ ❆ ❢✉♥❝t✐♦♥ f : V − → R ✐s ❝❛❧❧❡❞ ❛♥ ❡✐❣❡♥❢✉♥❝t✐♦♥ ♦❢ t❤❡ ❣r❛♣❤ G ❝♦rr❡s♣♦♥❞✐♥❣ t♦ ❛♥ ❡✐❣❡♥✈❛❧✉❡ λ ✐❢ f ≡ 0 ❛♥❞ ❢♦r ❛♥② ✈❡rt❡① x t❤❡ ❧♦❝❛❧ ❝♦♥❞✐t✐♦♥ λ · f (x) =
y∈N(x) f (y) ❤♦❧❞s✳
N(x) = {x1, . . . , xk} ✐s ♥❡✐❣❤❜♦r❤♦♦❞ λ · f (x) = f (x1) + f (x2) + . . . + f (xk)
❊❧❡♥❛ ❑♦♥st❛♥t✐♥♦✈❛ Pr♦♣❡rt✐❡s ♦❢ t❤❡ ❙t❛r ❣r❛♣❤s ❙❏❚❯✲✷✵✶✽ ✶✹ ✴ ✸✺
❚❤❡ ❝❛s❡ ♦❢ ❡✐❣❡♥✈❛❧✉❡ λ = n − 1 ✐s tr✐✈✐❛❧ ✇❤✐❝❤ ❢♦❧❧♦✇s ❢r♦♠ t❤❡ (n − 1)✲r❡❣✉❧❛r✐t② ♦❢ t❤❡ ❙t❛r ❣r❛♣❤✳ ✱
❊❧❡♥❛ ❑♦♥st❛♥t✐♥♦✈❛ Pr♦♣❡rt✐❡s ♦❢ t❤❡ ❙t❛r ❣r❛♣❤s ❙❏❚❯✲✷✵✶✽ ✶✺ ✴ ✸✺
❚❤❡ ❝❛s❡ ♦❢ ❡✐❣❡♥✈❛❧✉❡ λ = n − 1 ✐s tr✐✈✐❛❧ ✇❤✐❝❤ ❢♦❧❧♦✇s ❢r♦♠ t❤❡ (n − 1)✲r❡❣✉❧❛r✐t② ♦❢ t❤❡ ❙t❛r ❣r❛♣❤✳ x ∈ V (Sn)✱ N(x) = |T| = n − 1, (n − 1) · f (x) = 1 + 1 + . . . + 1
❊❧❡♥❛ ❑♦♥st❛♥t✐♥♦✈❛ Pr♦♣❡rt✐❡s ♦❢ t❤❡ ❙t❛r ❣r❛♣❤s ❙❏❚❯✲✷✵✶✽ ✶✺ ✴ ✸✺
❉❡✜♥❡ t❤❡ s❡t F2 = {f 2,k
i
| i ∈ {2, 3, . . . , n}, k ∈ {3, 4, . . . , n}}, ♦❢ t❤❡ PI✲❡✐❣❡♥❢✉♥❝t✐♦♥s✿ f j,k
i
(π) = 1, ✐❢ πj = i❀ −1, ✐❢ πk = i❀ 0, ♦t❤❡r✇✐s❡✱ ✇❤❡r❡ π = [π1π2 . . . πn].
❘❡s✉❧t ✶ ✭●♦r②❛✐♥♦✈✱ ❑❛❜❛♥♦✈✱ ❑✱ ❙❤❛❧❛❣✐♥♦✈✱ ❱❛❧②✉③❤❡♥✐❝❤✮
❚❤❡ s❡t ❢♦r♠s ❛ ❜❛s✐s ♦❢ t❤❡ ❡✐❣❡♥s♣❛❝❡ ♦❢ ❝♦rr❡s♣♦♥❞✐♥❣ t♦ t❤❡ ❧❛r❣❡st ♥♦♥✲♣r✐♥❝✐♣❛❧ ❡✐❣❡♥✈❛❧✉❡ ✳
❘❡s✉❧t ✷ ✭●❑❑❙❤❱✮
❆♥② ❡✐❣❡♥❢✉♥❝t✐♦♥ ♦❢ ✱ ✱ ✇✐t❤ ❡✐❣❡♥✈❛❧✉❡ ❝❛♥ ❜❡ ✉♥✐q✉❡❧② r❡❝♦♥str✉❝t❡❞ ❜② ✐ts ✈❛❧✉❡s ♦♥ t❤❡ s❡❝♦♥❞ ♥❡✐❣❤❜♦✉r❤♦♦❞ ♦❢ ❛♥② ✈❡rt❡①✳
❊❧❡♥❛ ❑♦♥st❛♥t✐♥♦✈❛ Pr♦♣❡rt✐❡s ♦❢ t❤❡ ❙t❛r ❣r❛♣❤s ❙❏❚❯✲✷✵✶✽ ✶✻ ✴ ✸✺
❉❡✜♥❡ t❤❡ s❡t F2 = {f 2,k
i
| i ∈ {2, 3, . . . , n}, k ∈ {3, 4, . . . , n}}, ♦❢ t❤❡ PI✲❡✐❣❡♥❢✉♥❝t✐♦♥s✿ f j,k
i
(π) = 1, ✐❢ πj = i❀ −1, ✐❢ πk = i❀ 0, ♦t❤❡r✇✐s❡✱ ✇❤❡r❡ π = [π1π2 . . . πn].
❘❡s✉❧t ✶ ✭●♦r②❛✐♥♦✈✱ ❑❛❜❛♥♦✈✱ ❑✱ ❙❤❛❧❛❣✐♥♦✈✱ ❱❛❧②✉③❤❡♥✐❝❤✮
❚❤❡ s❡t F2 ❢♦r♠s ❛ ❜❛s✐s ♦❢ t❤❡ ❡✐❣❡♥s♣❛❝❡ ♦❢ Sn ❝♦rr❡s♣♦♥❞✐♥❣ t♦ t❤❡ ❧❛r❣❡st ♥♦♥✲♣r✐♥❝✐♣❛❧ ❡✐❣❡♥✈❛❧✉❡ λ = n − 2✳
❘❡s✉❧t ✷ ✭●❑❑❙❤❱✮
❆♥② ❡✐❣❡♥❢✉♥❝t✐♦♥ ♦❢ ✱ ✱ ✇✐t❤ ❡✐❣❡♥✈❛❧✉❡ ❝❛♥ ❜❡ ✉♥✐q✉❡❧② r❡❝♦♥str✉❝t❡❞ ❜② ✐ts ✈❛❧✉❡s ♦♥ t❤❡ s❡❝♦♥❞ ♥❡✐❣❤❜♦✉r❤♦♦❞ ♦❢ ❛♥② ✈❡rt❡①✳
❊❧❡♥❛ ❑♦♥st❛♥t✐♥♦✈❛ Pr♦♣❡rt✐❡s ♦❢ t❤❡ ❙t❛r ❣r❛♣❤s ❙❏❚❯✲✷✵✶✽ ✶✻ ✴ ✸✺
❉❡✜♥❡ t❤❡ s❡t F2 = {f 2,k
i
| i ∈ {2, 3, . . . , n}, k ∈ {3, 4, . . . , n}}, ♦❢ t❤❡ PI✲❡✐❣❡♥❢✉♥❝t✐♦♥s✿ f j,k
i
(π) = 1, ✐❢ πj = i❀ −1, ✐❢ πk = i❀ 0, ♦t❤❡r✇✐s❡✱ ✇❤❡r❡ π = [π1π2 . . . πn].
❘❡s✉❧t ✶ ✭●♦r②❛✐♥♦✈✱ ❑❛❜❛♥♦✈✱ ❑✱ ❙❤❛❧❛❣✐♥♦✈✱ ❱❛❧②✉③❤❡♥✐❝❤✮
❚❤❡ s❡t F2 ❢♦r♠s ❛ ❜❛s✐s ♦❢ t❤❡ ❡✐❣❡♥s♣❛❝❡ ♦❢ Sn ❝♦rr❡s♣♦♥❞✐♥❣ t♦ t❤❡ ❧❛r❣❡st ♥♦♥✲♣r✐♥❝✐♣❛❧ ❡✐❣❡♥✈❛❧✉❡ λ = n − 2✳
❘❡s✉❧t ✷ ✭●❑❑❙❤❱✮
❆♥② ❡✐❣❡♥❢✉♥❝t✐♦♥ ♦❢ Sn✱ n 3✱ ✇✐t❤ ❡✐❣❡♥✈❛❧✉❡ n − 2 ❝❛♥ ❜❡ ✉♥✐q✉❡❧② r❡❝♦♥str✉❝t❡❞ ❜② ✐ts ✈❛❧✉❡s ♦♥ t❤❡ s❡❝♦♥❞ ♥❡✐❣❤❜♦✉r❤♦♦❞ ♦❢ ❛♥② ✈❡rt❡①✳
❊❧❡♥❛ ❑♦♥st❛♥t✐♥♦✈❛ Pr♦♣❡rt✐❡s ♦❢ t❤❡ ❙t❛r ❣r❛♣❤s ❙❏❚❯✲✷✵✶✽ ✶✻ ✴ ✸✺
❊❧❡♥❛ ❑♦♥st❛♥t✐♥♦✈❛ Pr♦♣❡rt✐❡s ♦❢ t❤❡ ❙t❛r ❣r❛♣❤s ❙❏❚❯✲✷✵✶✽ ✶✼ ✴ ✸✺
❊❧❡♥❛ ❑♦♥st❛♥t✐♥♦✈❛ Pr♦♣❡rt✐❡s ♦❢ t❤❡ ❙t❛r ❣r❛♣❤s ❙❏❚❯✲✷✵✶✽ ✶✽ ✴ ✸✺
1 1 1 1 1 1
❊❧❡♥❛ ❑♦♥st❛♥t✐♥♦✈❛ Pr♦♣❡rt✐❡s ♦❢ t❤❡ ❙t❛r ❣r❛♣❤s ❙❏❚❯✲✷✵✶✽ ✶✾ ✴ ✸✺
❲❡ ❞❡✜♥❡ ❛ ♠❛tr✐① Mn s✉❝❤ t❤❛t✿ r♦✇s ❛r❡ ✐♥❞❡①❡❞ ❜② ❛ s❡q✉❡♥❝❡ ♦❢ ❡✐❣❡♥❢✉♥❝t✐♦♥s ❢r♦♠ F2❀ ❝♦❧✉♠♥s ❛r❡ ✐♥❞❡①❡❞ ❜② ❛ s❡q✉❡♥❝❡ ♦❢ ♣❡r♠✉t❛t✐♦♥s ❢r♦♠ t❤❡ s❡❝♦♥❞ ♥❡✐❣❤❜♦✉r❤♦♦❞ ♦❢ t❤❡ ✐❞❡♥t✐t② ♣❡r♠✉t❛t✐♦♥ N2 = {(1rs) | r, s ∈ {2, . . . , n}, r = s}✱ |N2| = (n − 1)(n − 2)❀ t❤❡ ❡♥tr✐❡s ❛r❡ ✈❛❧✉❡s ♦❢ r♦✇ ❡✐❣❡♥❢✉♥❝t✐♦♥s ♦♥ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❝♦❧✉♠♥ ♣❡r♠✉t❛t✐♦♥s✳
❋❛❝t ✭●❑❑❙❤❱✮
❚❤❡ ❢♦❧❧♦✇✐♥❣ ❡q✉❛❧✐t② ❤♦❧❞s ■♥ ♣❛rt✐❝✉❧❛r✱ t❤❡ ♠❛tr✐① ✐s ♥♦♥✲❞❡❣❡♥❡r❛t❡✳
❊❧❡♥❛ ❑♦♥st❛♥t✐♥♦✈❛ Pr♦♣❡rt✐❡s ♦❢ t❤❡ ❙t❛r ❣r❛♣❤s ❙❏❚❯✲✷✵✶✽ ✷✵ ✴ ✸✺
❲❡ ❞❡✜♥❡ ❛ ♠❛tr✐① Mn s✉❝❤ t❤❛t✿ r♦✇s ❛r❡ ✐♥❞❡①❡❞ ❜② ❛ s❡q✉❡♥❝❡ ♦❢ ❡✐❣❡♥❢✉♥❝t✐♦♥s ❢r♦♠ F2❀ ❝♦❧✉♠♥s ❛r❡ ✐♥❞❡①❡❞ ❜② ❛ s❡q✉❡♥❝❡ ♦❢ ♣❡r♠✉t❛t✐♦♥s ❢r♦♠ t❤❡ s❡❝♦♥❞ ♥❡✐❣❤❜♦✉r❤♦♦❞ ♦❢ t❤❡ ✐❞❡♥t✐t② ♣❡r♠✉t❛t✐♦♥ N2 = {(1rs) | r, s ∈ {2, . . . , n}, r = s}✱ |N2| = (n − 1)(n − 2)❀ t❤❡ ❡♥tr✐❡s ❛r❡ ✈❛❧✉❡s ♦❢ r♦✇ ❡✐❣❡♥❢✉♥❝t✐♦♥s ♦♥ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❝♦❧✉♠♥ ♣❡r♠✉t❛t✐♦♥s✳
❋❛❝t ✭●❑❑❙❤❱✮
❚❤❡ ❢♦❧❧♦✇✐♥❣ ❡q✉❛❧✐t② ❤♦❧❞s det(Mn) = (−1)n−3(n − 2)n−2(n2 − 5n + 5). ■♥ ♣❛rt✐❝✉❧❛r✱ t❤❡ ♠❛tr✐① Mn ✐s ♥♦♥✲❞❡❣❡♥❡r❛t❡✳
❊❧❡♥❛ ❑♦♥st❛♥t✐♥♦✈❛ Pr♦♣❡rt✐❡s ♦❢ t❤❡ ❙t❛r ❣r❛♣❤s ❙❏❚❯✲✷✵✶✽ ✷✵ ✴ ✸✺
◆♦♥✲❞❡❣❡♥❡r❛❝② ♦❢ Mn ❣✐✈❡s ❘❡s✉❧t ✶
❚❤❡ s❡t F2 ❢♦r♠s ❛ ❜❛s✐s ♦❢ t❤❡ ❡✐❣❡♥s♣❛❝❡ ♦❢ Sn ❝♦rr❡s♣♦♥❞✐♥❣ t♦ t❤❡ ❧❛r❣❡st ♥♦♥✲♣r✐♥❝✐♣❛❧ ❡✐❣❡♥✈❛❧✉❡ λ = n − 2✳ ❙✐♥❝❡ t❤❡ r♦✇s ♦❢ t❤❡ ♠❛tr✐① ❛r❡ t❤❡ r❡str✐❝t✐♦♥s ♦❢ t❤❡ ✲❡✐❣❡♥❢✉♥❝t✐♦♥s ❢r♦♠ ♦♥ t❤❡ s❡t ✱ ✇❤❡r❡ ❢r♦♠ ♥♦♥✲❞❡❣❡♥❡r❛❝② ♦❢ ✱ ❘❡s✉❧t ✶ ❛♥❞ ✈❡rt❡①✲tr❛♥s✐t✐✈✐t② ♦❢ ✇❡ ❤❛✈❡ ❘❡s✉❧t ✷✳
❘❡s✉❧t ✷ ✭●❑❑❙❤❱✮
❆♥② ❡✐❣❡♥❢✉♥❝t✐♦♥ ♦❢ ✱ ✱ ✇✐t❤ ❡✐❣❡♥✈❛❧✉❡ ❝❛♥ ❜❡ ✉♥✐q✉❡❧② r❡❝♦♥str✉❝t❡❞ ❜② ✐ts ✈❛❧✉❡s ♦♥ t❤❡ s❡❝♦♥❞ ♥❡✐❣❤❜♦✉r❤♦♦❞ ♦❢ ❛♥② ✈❡rt❡①✳
❊❧❡♥❛ ❑♦♥st❛♥t✐♥♦✈❛ Pr♦♣❡rt✐❡s ♦❢ t❤❡ ❙t❛r ❣r❛♣❤s ❙❏❚❯✲✷✵✶✽ ✷✶ ✴ ✸✺
◆♦♥✲❞❡❣❡♥❡r❛❝② ♦❢ Mn ❣✐✈❡s ❘❡s✉❧t ✶
❚❤❡ s❡t F2 ❢♦r♠s ❛ ❜❛s✐s ♦❢ t❤❡ ❡✐❣❡♥s♣❛❝❡ ♦❢ Sn ❝♦rr❡s♣♦♥❞✐♥❣ t♦ t❤❡ ❧❛r❣❡st ♥♦♥✲♣r✐♥❝✐♣❛❧ ❡✐❣❡♥✈❛❧✉❡ λ = n − 2✳ ❙✐♥❝❡ t❤❡ r♦✇s ♦❢ t❤❡ ♠❛tr✐① Mn ❛r❡ t❤❡ r❡str✐❝t✐♦♥s ♦❢ t❤❡ PI✲❡✐❣❡♥❢✉♥❝t✐♦♥s ❢r♦♠ F2 ♦♥ t❤❡ s❡t N2✱ ✇❤❡r❡ |N2| = |F2| = (n − 1)(n − 2), ❢r♦♠ ♥♦♥✲❞❡❣❡♥❡r❛❝② ♦❢ Mn✱ ❘❡s✉❧t ✶ ❛♥❞ ✈❡rt❡①✲tr❛♥s✐t✐✈✐t② ♦❢ Sn ✇❡ ❤❛✈❡ ❘❡s✉❧t ✷✳
❘❡s✉❧t ✷ ✭●❑❑❙❤❱✮
❆♥② ❡✐❣❡♥❢✉♥❝t✐♦♥ ♦❢ ✱ ✱ ✇✐t❤ ❡✐❣❡♥✈❛❧✉❡ ❝❛♥ ❜❡ ✉♥✐q✉❡❧② r❡❝♦♥str✉❝t❡❞ ❜② ✐ts ✈❛❧✉❡s ♦♥ t❤❡ s❡❝♦♥❞ ♥❡✐❣❤❜♦✉r❤♦♦❞ ♦❢ ❛♥② ✈❡rt❡①✳
❊❧❡♥❛ ❑♦♥st❛♥t✐♥♦✈❛ Pr♦♣❡rt✐❡s ♦❢ t❤❡ ❙t❛r ❣r❛♣❤s ❙❏❚❯✲✷✵✶✽ ✷✶ ✴ ✸✺
◆♦♥✲❞❡❣❡♥❡r❛❝② ♦❢ Mn ❣✐✈❡s ❘❡s✉❧t ✶
❚❤❡ s❡t F2 ❢♦r♠s ❛ ❜❛s✐s ♦❢ t❤❡ ❡✐❣❡♥s♣❛❝❡ ♦❢ Sn ❝♦rr❡s♣♦♥❞✐♥❣ t♦ t❤❡ ❧❛r❣❡st ♥♦♥✲♣r✐♥❝✐♣❛❧ ❡✐❣❡♥✈❛❧✉❡ λ = n − 2✳ ❙✐♥❝❡ t❤❡ r♦✇s ♦❢ t❤❡ ♠❛tr✐① Mn ❛r❡ t❤❡ r❡str✐❝t✐♦♥s ♦❢ t❤❡ PI✲❡✐❣❡♥❢✉♥❝t✐♦♥s ❢r♦♠ F2 ♦♥ t❤❡ s❡t N2✱ ✇❤❡r❡ |N2| = |F2| = (n − 1)(n − 2), ❢r♦♠ ♥♦♥✲❞❡❣❡♥❡r❛❝② ♦❢ Mn✱ ❘❡s✉❧t ✶ ❛♥❞ ✈❡rt❡①✲tr❛♥s✐t✐✈✐t② ♦❢ Sn ✇❡ ❤❛✈❡ ❘❡s✉❧t ✷✳
❘❡s✉❧t ✷ ✭●❑❑❙❤❱✮
❆♥② ❡✐❣❡♥❢✉♥❝t✐♦♥ ♦❢ Sn✱ n 3✱ ✇✐t❤ ❡✐❣❡♥✈❛❧✉❡ n − 2 ❝❛♥ ❜❡ ✉♥✐q✉❡❧② r❡❝♦♥str✉❝t❡❞ ❜② ✐ts ✈❛❧✉❡s ♦♥ t❤❡ s❡❝♦♥❞ ♥❡✐❣❤❜♦✉r❤♦♦❞ ♦❢ ❛♥② ✈❡rt❡①✳
❊❧❡♥❛ ❑♦♥st❛♥t✐♥♦✈❛ Pr♦♣❡rt✐❡s ♦❢ t❤❡ ❙t❛r ❣r❛♣❤s ❙❏❚❯✲✷✵✶✽ ✷✶ ✴ ✸✺
❚❤❡ ❙t❛r ❣r❛♣❤ Sn = Cay(Symn, T), n 2
✐s t❤❡ ❈❛②❧❡② ❣r❛♣❤ ♦♥ t❤❡ s②♠♠❡tr✐❝ ❣r♦✉♣ Symn ♦❢ ♣❡r♠✉t❛t✐♦♥s π = [π1π2...πi...πn] ✇✐t❤ t❤❡ ❣❡♥❡r❛t✐♥❣ s❡t T ♦❢ ❛❧❧ tr❛♥s♣♦s✐t✐♦♥s ti = (1 i) s✇❛♣♣✐♥❣ t❤❡ 1st ❛♥❞ it❤ ❡❧❡♠❡♥ts ♦❢ ❛ ♣❡r♠✉t❛t✐♦♥ π✳
Pr♦♣❡rt✐❡s ♦❢ t❤❡ ❙t❛r ❣r❛♣❤
❝♦♥♥❡❝t❡❞ ❜✐♣❛rt✐t❡ (n − 1)✕r❡❣✉❧❛r ❣r❛♣❤ ♦❢ ♦r❞❡r n! ❛♥❞ ❞✐❛♠❡t❡r diam(Sn) = ⌊ 3(n−1)
2
⌋ ✭❙✳ ❇✳ ❆❦❡rs✱ ❇✳ ❑r✐s❤♥❛♠✉rt❤②✱ 1989✮ ✈❡rt❡①✲tr❛♥s✐t✐✈❡ ❛♥❞ ❡❞❣❡✲tr❛♥s✐t✐✈❡ ❝♦♥t❛✐♥s ❤❛♠✐❧t♦♥✐❛♥ ❝②❝❧❡s ✭❱✳ ❑♦♠♣❡❧✬♠❛❦❤❡r✱ ❱✳ ▲✐s❦♦✈❡ts✱ 1975✱ P✳ ❙❧❛t❡r✱ 1978✮ ✐t ❞♦❡s ❝♦♥t❛✐♥ ❡✈❡♥ ℓ✕❝②❝❧❡s ✇❤❡r❡ ℓ = 6, 8, . . . , n! ⇒ ❍❛♠✐❧t♦♥✐❛♥ ❤❛s ❤✐❡r❛r❝❤✐❝❛❧ str✉❝t✉r❡ ❤❛s ✐♥t❡❣r❛❧ s♣❡❝tr✉♠
❊❧❡♥❛ ❑♦♥st❛♥t✐♥♦✈❛ Pr♦♣❡rt✐❡s ♦❢ t❤❡ ❙t❛r ❣r❛♣❤s ❙❏❚❯✲✷✵✶✽ ✷✷ ✴ ✸✺
❆✳ ❲✐❧❧✐❛♠s✱ ❏✳ ❙❛✇❛❞❛✱ ●r❡❡❞② ♣❛♥❝❛❦❡ ✢✐♣♣✐♥❣ ✭✷✵✶✸✮
❚❛❦❡ ❛ st❛❝❦ ♦❢ ♣❛♥❝❛❦❡s✱ ♥✉♠❜❡r❡❞ 1, 2, . . . , n ❜② ✐♥❝r❡❛s✐♥❣ ❞✐❛♠❡t❡r✱ ❛♥❞ r❡♣❡❛t t❤❡ ❢♦❧❧♦✇✐♥❣✿ ❋❧✐♣ t❤❡ ♠❛①✐♠✉♠ ♥✉♠❜❡r ♦❢ t♦♣♠♦st ♣❛♥❝❛❦❡s t❤❛t ❣✐✈❡s ❛ ♥❡✇ st❛❝❦✳
❊❧❡♥❛ ❑♦♥st❛♥t✐♥♦✈❛ Pr♦♣❡rt✐❡s ♦❢ t❤❡ ❙t❛r ❣r❛♣❤s ❙❏❚❯✲✷✵✶✽ ✷✸ ✴ ✸✺
❆✳ ❲✐❧❧✐❛♠s✱ ❏✳ ❙❛✇❛❞❛✱ ●r❡❡❞② ♣❛♥❝❛❦❡ ✢✐♣♣✐♥❣ ✭✷✵✶✸✮
❚❛❦❡ ❛ st❛❝❦ ♦❢ ♣❛♥❝❛❦❡s✱ ♥✉♠❜❡r❡❞ 1, 2, . . . , n ❜② ✐♥❝r❡❛s✐♥❣ ❞✐❛♠❡t❡r✱ ❛♥❞ r❡♣❡❛t t❤❡ ❢♦❧❧♦✇✐♥❣✿ ❋❧✐♣ t❤❡ ♠❛①✐♠✉♠ ♥✉♠❜❡r ♦❢ t♦♣♠♦st ♣❛♥❝❛❦❡s t❤❛t ❣✐✈❡s ❛ ♥❡✇ st❛❝❦✳ [1234] [4321] [2341] [1432] [3412] [2143] [4123] [3214] [2314] [4132] [3142] [2413] [1423] [3241] [4231] [1324] [3124] [4213] [1243] [3421] [2431] [1342] [4312] [2134]
❊❧❡♥❛ ❑♦♥st❛♥t✐♥♦✈❛ Pr♦♣❡rt✐❡s ♦❢ t❤❡ ❙t❛r ❣r❛♣❤s ❙❏❚❯✲✷✵✶✽ ✷✸ ✴ ✸✺
❊❛❝❤ ✬✢✐♣✬ ✐s ❢♦r♠❛❧❧② ❦♥♦✇♥ ❛s ♣r❡✜①✕r❡✈❡rs❛❧✳
❚❤❡ P❛♥❝❛❦❡ ❣r❛♣❤
✐s t❤❡ ❈❛②❧❡② ❣r❛♣❤ ♦♥ t❤❡ s②♠♠❡tr✐❝ ❣r♦✉♣ ✇✐t❤ ❣❡♥❡r❛t✐♥❣ s❡t ✇❤❡r❡ ✐s t❤❡ ♦♣❡r❛t✐♦♥ ♦❢ r❡✈❡rs✐♥❣ t❤❡ ♦r❞❡r ♦❢ ❛♥② s✉❜str✐♥❣ ♦❢ ❛ ♣❡r♠✉t❛t✐♦♥ ✇❤❡♥ ♠✉❧t✐♣❧✐❡❞ ♦♥ t❤❡ r✐❣❤t✱ ✐✳❡✳✱ ✳
❲✐❧❧✐❛♠s✬ ♣r❡✜①✕r❡✈❡rs❛❧ ●r❛② ❝♦❞❡✿
❋❧✐♣ t❤❡ ♠❛①✐♠✉♠ ♥✉♠❜❡r ♦❢ t♦♣♠♦st ♣❛♥❝❛❦❡s t❤❛t ❣✐✈❡s ❛ ♥❡✇ st❛❝❦✳
❩❛❦s✬ ✭✶✾✽✹✮ ♣r❡✜①✕r❡✈❡rs❛❧ ●r❛② ❝♦❞❡✿
❋❧✐♣ t❤❡ ♠✐♥✐♠✉♠ ♥✉♠❜❡r ♦❢ t♦♣♠♦st ♣❛♥❝❛❦❡s t❤❛t ❣✐✈❡s ❛ ♥❡✇ st❛❝❦✳
❊❧❡♥❛ ❑♦♥st❛♥t✐♥♦✈❛ Pr♦♣❡rt✐❡s ♦❢ t❤❡ ❙t❛r ❣r❛♣❤s ❙❏❚❯✲✷✵✶✽ ✷✹ ✴ ✸✺
❊❛❝❤ ✬✢✐♣✬ ✐s ❢♦r♠❛❧❧② ❦♥♦✇♥ ❛s ♣r❡✜①✕r❡✈❡rs❛❧✳
❚❤❡ P❛♥❝❛❦❡ ❣r❛♣❤ Sn = Cay(Symn, PR), n 2
✐s t❤❡ ❈❛②❧❡② ❣r❛♣❤ ♦♥ t❤❡ s②♠♠❡tr✐❝ ❣r♦✉♣ Symn ✇✐t❤ ❣❡♥❡r❛t✐♥❣ s❡t {ri ∈ Symn, 1 i < n}, ✇❤❡r❡ ri ✐s t❤❡ ♦♣❡r❛t✐♦♥ ♦❢ r❡✈❡rs✐♥❣ t❤❡ ♦r❞❡r ♦❢ ❛♥② s✉❜str✐♥❣ [1, i], 1 < i n, ♦❢ ❛ ♣❡r♠✉t❛t✐♦♥ π ✇❤❡♥ ♠✉❧t✐♣❧✐❡❞ ♦♥ t❤❡ r✐❣❤t✱ ✐✳❡✳✱ [π1 . . . πiπi+1 . . . πn]ri = [πi . . . π1πi+1 . . . πn]✳
❲✐❧❧✐❛♠s✬ ♣r❡✜①✕r❡✈❡rs❛❧ ●r❛② ❝♦❞❡✿
❋❧✐♣ t❤❡ ♠❛①✐♠✉♠ ♥✉♠❜❡r ♦❢ t♦♣♠♦st ♣❛♥❝❛❦❡s t❤❛t ❣✐✈❡s ❛ ♥❡✇ st❛❝❦✳
❩❛❦s✬ ✭✶✾✽✹✮ ♣r❡✜①✕r❡✈❡rs❛❧ ●r❛② ❝♦❞❡✿
❋❧✐♣ t❤❡ ♠✐♥✐♠✉♠ ♥✉♠❜❡r ♦❢ t♦♣♠♦st ♣❛♥❝❛❦❡s t❤❛t ❣✐✈❡s ❛ ♥❡✇ st❛❝❦✳
❊❧❡♥❛ ❑♦♥st❛♥t✐♥♦✈❛ Pr♦♣❡rt✐❡s ♦❢ t❤❡ ❙t❛r ❣r❛♣❤s ❙❏❚❯✲✷✵✶✽ ✷✹ ✴ ✸✺
❊❛❝❤ ✬✢✐♣✬ ✐s ❢♦r♠❛❧❧② ❦♥♦✇♥ ❛s ♣r❡✜①✕r❡✈❡rs❛❧✳
❚❤❡ P❛♥❝❛❦❡ ❣r❛♣❤ Sn = Cay(Symn, PR), n 2
✐s t❤❡ ❈❛②❧❡② ❣r❛♣❤ ♦♥ t❤❡ s②♠♠❡tr✐❝ ❣r♦✉♣ Symn ✇✐t❤ ❣❡♥❡r❛t✐♥❣ s❡t {ri ∈ Symn, 1 i < n}, ✇❤❡r❡ ri ✐s t❤❡ ♦♣❡r❛t✐♦♥ ♦❢ r❡✈❡rs✐♥❣ t❤❡ ♦r❞❡r ♦❢ ❛♥② s✉❜str✐♥❣ [1, i], 1 < i n, ♦❢ ❛ ♣❡r♠✉t❛t✐♦♥ π ✇❤❡♥ ♠✉❧t✐♣❧✐❡❞ ♦♥ t❤❡ r✐❣❤t✱ ✐✳❡✳✱ [π1 . . . πiπi+1 . . . πn]ri = [πi . . . π1πi+1 . . . πn]✳
❲✐❧❧✐❛♠s✬ ♣r❡✜①✕r❡✈❡rs❛❧ ●r❛② ❝♦❞❡✿ rn rn−1 rn−2, . . . , r3, r2
❋❧✐♣ t❤❡ ♠❛①✐♠✉♠ ♥✉♠❜❡r ♦❢ t♦♣♠♦st ♣❛♥❝❛❦❡s t❤❛t ❣✐✈❡s ❛ ♥❡✇ st❛❝❦✳
❩❛❦s✬ ✭✶✾✽✹✮ ♣r❡✜①✕r❡✈❡rs❛❧ ●r❛② ❝♦❞❡✿ r2 r3, . . . , rn−2 rn−1 rn
❋❧✐♣ t❤❡ ♠✐♥✐♠✉♠ ♥✉♠❜❡r ♦❢ t♦♣♠♦st ♣❛♥❝❛❦❡s t❤❛t ❣✐✈❡s ❛ ♥❡✇ st❛❝❦✳
❊❧❡♥❛ ❑♦♥st❛♥t✐♥♦✈❛ Pr♦♣❡rt✐❡s ♦❢ t❤❡ ❙t❛r ❣r❛♣❤s ❙❏❚❯✲✷✵✶✽ ✷✹ ✴ ✸✺
✭r4, r3, r2✮
❊❧❡♥❛ ❑♦♥st❛♥t✐♥♦✈❛ Pr♦♣❡rt✐❡s ♦❢ t❤❡ ❙t❛r ❣r❛♣❤s ❙❏❚❯✲✷✵✶✽ ✷✺ ✴ ✸✺
❆r❡ t❤❡r❡ ❣r❡❡❞② ❤❛♠✐❧t♦♥✐❛♥ ❝②❝❧❡s ✐♥ ♦t❤❡r ❈❛②❧❡② ❣r❛♣❤s❄
◗✉❡st✐♦♥
❆r❡ t❤❡r❡ ❣r❡❡❞② ❤❛♠✐❧t♦♥✐❛♥ ❝②❝❧❡s ✐♥ t❤❡ ❙t❛r ❣r❛♣❤❄
❊❧❡♥❛ ❑♦♥st❛♥t✐♥♦✈❛ Pr♦♣❡rt✐❡s ♦❢ t❤❡ ❙t❛r ❣r❛♣❤s ❙❏❚❯✲✷✵✶✽ ✷✻ ✴ ✸✺
❆r❡ t❤❡r❡ ❣r❡❡❞② ❤❛♠✐❧t♦♥✐❛♥ ❝②❝❧❡s ✐♥ ♦t❤❡r ❈❛②❧❡② ❣r❛♣❤s❄
◗✉❡st✐♦♥
❆r❡ t❤❡r❡ ❣r❡❡❞② ❤❛♠✐❧t♦♥✐❛♥ ❝②❝❧❡s ✐♥ t❤❡ ❙t❛r ❣r❛♣❤❄
❊❧❡♥❛ ❑♦♥st❛♥t✐♥♦✈❛ Pr♦♣❡rt✐❡s ♦❢ t❤❡ ❙t❛r ❣r❛♣❤s ❙❏❚❯✲✷✵✶✽ ✷✻ ✴ ✸✺
[1234] [2134] [3124] [1324] [2314] [3214] [4213] [2413] [1423] [4123] [2143] [1243] [3241] [2341] [4321] [3421] [2431] [4231]
✲
t2
✲
t3
✲
t2
✲
t3
✲
t2
✲
t4
✲
t2
✲
t3
✲
t2
✲
t3
✲
t2
✲
t4
✲
t2
✲
t3
✲
t2
✲
t3
✲
t2
❊❧❡♥❛ ❑♦♥st❛♥t✐♥♦✈❛ Pr♦♣❡rt✐❡s ♦❢ t❤❡ ❙t❛r ❣r❛♣❤s ❙❏❚❯✲✷✵✶✽ ✷✼ ✴ ✸✺
❚❤❡♦r❡♠ ✭❉✳ ●♦st❡✈s❦②✱ ❊❑✱ 2018✮
■♥ t❤❡ ❙t❛r ❣r❛♣❤ Sn, n 3✱ ❛♥② ❣r❡❡❞② s❡q✉❡♥❝❡ GS ♦❢ ❧❡♥❣t❤ k✱ ✇❤❡r❡ 2 k n − 1✱ ❢♦r♠s ❛ GS✲❣r❡❡❞② ❝②❝❧❡ ♦❢ ❧❡♥❣t❤ 2 · 3k−1✳ Pr♦♦❢ ■❢ ✱ t❤❡♥ ✱ ❤❡♥❝❡ ✐s ❛ ❣r❡❡❞② s❡q✉❡♥❝❡ ❣❡♥❡r❛t✐♥❣ s✐① ♣❡r♠✉t❛t✐♦♥s ❛s ❢♦❧❧♦✇s✿ ✇❤✐❝❤ ♦❜✈✐♦✉s❧② ❢♦r♠s ❛ ❝②❝❧❡ ♦❢ ❧❡♥❣t❤ ✳
❊❧❡♥❛ ❑♦♥st❛♥t✐♥♦✈❛ Pr♦♣❡rt✐❡s ♦❢ t❤❡ ❙t❛r ❣r❛♣❤s ❙❏❚❯✲✷✵✶✽ ✷✽ ✴ ✸✺
❚❤❡♦r❡♠ ✭❉✳ ●♦st❡✈s❦②✱ ❊❑✱ 2018✮
■♥ t❤❡ ❙t❛r ❣r❛♣❤ Sn, n 3✱ ❛♥② ❣r❡❡❞② s❡q✉❡♥❝❡ GS ♦❢ ❧❡♥❣t❤ k✱ ✇❤❡r❡ 2 k n − 1✱ ❢♦r♠s ❛ GS✲❣r❡❡❞② ❝②❝❧❡ ♦❢ ❧❡♥❣t❤ 2 · 3k−1✳ Pr♦♦❢ ■❢ n = 3✱ t❤❡♥ S3 ∼ = C6✱ ❤❡♥❝❡ GS3 = (t2, t3) ✐s ❛ ❣r❡❡❞② s❡q✉❡♥❝❡ ❣❡♥❡r❛t✐♥❣ s✐① ♣❡r♠✉t❛t✐♦♥s ❛s ❢♦❧❧♦✇s✿ GS3 : [123] [213] [312] [132] [231] [321],
✲
t2
✲
t3
✲
t2
✲
t3
✲
t2
✇❤✐❝❤ ♦❜✈✐♦✉s❧② ❢♦r♠s ❛ ❝②❝❧❡ ♦❢ ❧❡♥❣t❤ 2 · 32−1 = 6✳
❊❧❡♥❛ ❑♦♥st❛♥t✐♥♦✈❛ Pr♦♣❡rt✐❡s ♦❢ t❤❡ ❙t❛r ❣r❛♣❤s ❙❏❚❯✲✷✵✶✽ ✷✽ ✴ ✸✺
❚❤❡♦r❡♠ ✭❉✳ ●♦st❡✈s❦②✱ ❊❑✱ 2018✮
■♥ t❤❡ ❙t❛r ❣r❛♣❤ Sn, n 3✱ ❛♥② ❣r❡❡❞② s❡q✉❡♥❝❡ GS ♦❢ ❧❡♥❣t❤ k✱ ✇❤❡r❡ 2 k n − 1✱ ❢♦r♠s ❛ GS✲❣r❡❡❞② ❝②❝❧❡ ♦❢ ❧❡♥❣t❤ 2 · 3k−1✳ Pr♦♦❢ ■❢ n = 4✱ t❤❡♥ GS4 = (t2, t3, t4) ❢♦r♠s ❛ ❣r❡❡❞② ❝②❝❧❡ ♦❢ ❧❡♥❣t❤ 6 · 3 = 2 · 33−1 = 18 ✐♥ S4✿ [1234] [2134] [3124] [1324] [2314] [3214] [4213] [2413] [1423] [4123] [2143] [1243] [3241] [2341] [4321] [3421] [2431] [4231].
✲
t2
✲
t3
✲
t2
✲
t3
✲
t2
✲
t4
✲
t2
✲
t3
✲
t2
✲
t3
✲
t2
✲
t4
✲
t2
✲
t3
✲
t2
✲
t3
✲
t2
❊❧❡♥❛ ❑♦♥st❛♥t✐♥♦✈❛ Pr♦♣❡rt✐❡s ♦❢ t❤❡ ❙t❛r ❣r❛♣❤s ❙❏❚❯✲✷✵✶✽ ✷✾ ✴ ✸✺
❚❤❡♦r❡♠ ✭❉✳ ●♦st❡✈s❦②✱ ❊❑✱ 2018✮
■♥ t❤❡ ❙t❛r ❣r❛♣❤ Sn, n 3✱ ❛♥② ❣r❡❡❞② s❡q✉❡♥❝❡ GS ♦❢ ❧❡♥❣t❤ k✱ ✇❤❡r❡ 2 k n − 1✱ ❢♦r♠s ❛ GS✲❣r❡❡❞② ❝②❝❧❡ ♦❢ ❧❡♥❣t❤ 2 · 3k−1✳ Pr♦♦❢ ❈♦♥s✐❞❡r ❛ s❡q✉❡♥❝❡ GSn = (t2, t3, t4, . . . , tn)
[1 2 3 . . . n − 1 n] [n − 1 2 3 . . . 1 n] ✲
GSn−1
✲
tn
[n 2 3 . . . 1 n − 1] [1 2 3 . . . n n − 1] ✲
GSn−1
✲
tn
[n − 1 2 3 . . . n 1] [n 2 3 . . . n − 1 1] ✲
GSn−1
❈♦r♦❧❧❛r②
❚❤❡r❡ ❛r❡ ♥♦ ❣r❡❡❞② ❤❛♠✐❧t♦♥✐❛♥ ❝②❝❧❡s ✐♥ ❢♦r ✳
❊❧❡♥❛ ❑♦♥st❛♥t✐♥♦✈❛ Pr♦♣❡rt✐❡s ♦❢ t❤❡ ❙t❛r ❣r❛♣❤s ❙❏❚❯✲✷✵✶✽ ✸✵ ✴ ✸✺
❚❤❡♦r❡♠ ✭❉✳ ●♦st❡✈s❦②✱ ❊❑✱ 2018✮
■♥ t❤❡ ❙t❛r ❣r❛♣❤ Sn, n 3✱ ❛♥② ❣r❡❡❞② s❡q✉❡♥❝❡ GS ♦❢ ❧❡♥❣t❤ k✱ ✇❤❡r❡ 2 k n − 1✱ ❢♦r♠s ❛ GS✲❣r❡❡❞② ❝②❝❧❡ ♦❢ ❧❡♥❣t❤ 2 · 3k−1✳ Pr♦♦❢ ❈♦♥s✐❞❡r ❛ s❡q✉❡♥❝❡ GSn = (t2, t3, t4, . . . , tn)
[1 2 3 . . . n − 1 n] [n − 1 2 3 . . . 1 n] ✲
GSn−1
✲
tn
[n 2 3 . . . 1 n − 1] [1 2 3 . . . n n − 1] ✲
GSn−1
✲
tn
[n − 1 2 3 . . . n 1] [n 2 3 . . . n − 1 1] ✲
GSn−1
❈♦r♦❧❧❛r②
❚❤❡r❡ ❛r❡ ♥♦ ❣r❡❡❞② ❤❛♠✐❧t♦♥✐❛♥ ❝②❝❧❡s ✐♥ Sn ❢♦r n 4✳
❊❧❡♥❛ ❑♦♥st❛♥t✐♥♦✈❛ Pr♦♣❡rt✐❡s ♦❢ t❤❡ ❙t❛r ❣r❛♣❤s ❙❏❚❯✲✷✵✶✽ ✸✵ ✴ ✸✺
▲❡t F = {GSk = (t2, t3, . . . , tk), 3 k n} ❜❡ ❛ ❢❛♠✐❧② ♦❢ ❣r❡❡❞② s❡q✉❡♥❝❡s✳
❚❤❡♦r❡♠ ✭❉✳ ●♦st❡✈s❦②✱ ❊❑✱ ✮
■♥ t❤❡ ❙t❛r ❣r❛♣❤ ✱ t❤❡r❡ ❡①✐sts ❛ ♠❛①✐♠❛❧ s❡t ♦❢ ✐♥❞❡♣❡♥❞❡♥t ❝②❝❧❡s ❢♦r♠❡❞ ❜② ❣r❡❡❞② s❡q✉❡♥❝❡s ❢r♦♠ t❤❡ ❢❛♠✐❧② ❝♦♥s✐st✐♥❣ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝②❝❧❡s✿ ✭✶✮ ♦♥❡ ❝②❝❧❡ ♦❢ ❧❡♥❣t❤ ✱ ❛♥❞ ✭✷✮ ❝②❝❧❡s ♦❢ ❧❡♥❣t❤ ✇❤❡♥ ✱ ❛♥❞ ✭✸✮ ❝②❝❧❡s ♦❢ ❧❡♥❣t❤ ❢♦r ❛❧❧ ✇❤❡♥ ✱ ✇❤❡r❡
❊❧❡♥❛ ❑♦♥st❛♥t✐♥♦✈❛ Pr♦♣❡rt✐❡s ♦❢ t❤❡ ❙t❛r ❣r❛♣❤s ❙❏❚❯✲✷✵✶✽ ✸✶ ✴ ✸✺
▲❡t F = {GSk = (t2, t3, . . . , tk), 3 k n} ❜❡ ❛ ❢❛♠✐❧② ♦❢ ❣r❡❡❞② s❡q✉❡♥❝❡s✳
❚❤❡♦r❡♠ ✭❉✳ ●♦st❡✈s❦②✱ ❊❑✱ 2018✮
■♥ t❤❡ ❙t❛r ❣r❛♣❤ Sn, n 3✱ t❤❡r❡ ❡①✐sts ❛ ♠❛①✐♠❛❧ s❡t ♦❢ ✐♥❞❡♣❡♥❞❡♥t ❝②❝❧❡s ❢♦r♠❡❞ ❜② ❣r❡❡❞② s❡q✉❡♥❝❡s ❢r♦♠ t❤❡ ❢❛♠✐❧② F ❝♦♥s✐st✐♥❣ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝②❝❧❡s✿ ✭✶✮ ♦♥❡ ❝②❝❧❡ ♦❢ ❧❡♥❣t❤ 2 · 3n−2✱ ❛♥❞ ✭✷✮ n − 3 ❝②❝❧❡s ♦❢ ❧❡♥❣t❤ 2 · 3n−3 ✇❤❡♥ n 4✱ ❛♥❞ ✭✸✮ Nm ❝②❝❧❡s ♦❢ ❧❡♥❣t❤ 2 · 3n−m−2 ❢♦r ❛❧❧ 2 m n − 3 ✇❤❡♥ n 5✱ ✇❤❡r❡ Nm = m
(n − l + 2)
❊❧❡♥❛ ❑♦♥st❛♥t✐♥♦✈❛ Pr♦♣❡rt✐❡s ♦❢ t❤❡ ❙t❛r ❣r❛♣❤s ❙❏❚❯✲✷✵✶✽ ✸✶ ✴ ✸✺
■♥❞❡♣❡♥❞❡♥t ❣r❡❡❞② 18✲ ❛♥❞ 6✲❝②❝❧❡s ❛r❡ ❢♦r♠❡❞ ❜② ❣r❡❡❞② s❡q✉❡♥❝❡s GS4 = (t2, t3, t4) ❛♥❞ GS3 = (t2, t3)✳
❊❧❡♥❛ ❑♦♥st❛♥t✐♥♦✈❛ Pr♦♣❡rt✐❡s ♦❢ t❤❡ ❙t❛r ❣r❛♣❤s ❙❏❚❯✲✷✵✶✽ ✸✷ ✴ ✸✺
❚❤❡r❡ ❛r❡ ❦♥♦✇♥ r❡s✉❧ts ❢♦r ❈❛②❧❡② ❣r❛♣❤s ♦♥ t❤❡ s②♠♠❡tr✐❝ ❣r♦✉♣ Symn ❣❡♥❡r❛t❡❞ ❜② tr❛♥s♣♦s✐t✐♦♥s✳
❱✳▲✳ ❑♦♠♣❡❧✬♠❛❦❤❡r✱ ❱✳❆✳ ▲✐s❦♦✈❡ts✱ ❙✉❝❝❡ss✐✈❡ ❣❡♥❡r❛t✐♦♥ ♦❢ ♣❡r♠✉t❛t✐♦♥s ❜② ♠❡❛♥s ♦❢ ❛ tr❛♥s♣♦s✐t✐♦♥ ❜❛s✐s (1975)
❚❤❡ ❣r❛♣❤ Cay(Symn, S) ✐s ❍❛♠✐❧t♦♥✐❛♥ ✇❤❡♥❡✈❡r S ✐s ❛ ❣❡♥❡r❛t✐♥❣ s❡t ❢♦r Symn ❝♦♥s✐st✐♥❣ ♦❢ tr❛♥s♣♦s✐t✐♦♥s✳
▼✳ ❚❝❤✉❡♥t❡✱ ●❡♥❡r❛t✐♦♥ ♦❢ ♣❡r♠✉t❛t✐♦♥s ❜② ❣r❛♣❤✐❝❛❧ ❡①❝❤❛♥❣❡s (1982)
▲❡t S ❜❡ ❛ s❡t ♦❢ tr❛♥s♣♦s✐t✐♦♥s t❤❛t ❣❡♥❡r❛t❡ Symn✳ ❚❤❡♥ t❤❡r❡ ✐s ❛ ❍❛♠✐❧t♦♥✐❛♥ ♣❛t❤ ✐♥ t❤❡ ❣r❛♣❤ Cay(Symn, S) ❥♦✐♥✐♥❣ ❛♥② ♣❡r♠✉t❛t✐♦♥s ♦❢ ♦♣♣♦s✐t❡ ♣❛r✐t②✳ ❚❤✉s✱ ❛❧❧ tr❛♥s♣♦s✐t✐♦♥ ❈❛②❧❡② ❣r❛♣❤s ❛r❡ ❍❛♠✐❧t♦♥✐❛♥✳
❊❧❡♥❛ ❑♦♥st❛♥t✐♥♦✈❛ Pr♦♣❡rt✐❡s ♦❢ t❤❡ ❙t❛r ❣r❛♣❤s ❙❏❚❯✲✷✵✶✽ ✸✸ ✴ ✸✺
❚❤❡r❡ ❛r❡ ♦♥❧② 4 ✈❡rt❡①✕tr❛♥s✐t✐✈❡ ✭♥♦t ❈❛②❧❡②✮ ❣r❛♣❤s ✇❤✐❝❤ ❞♦ ♥♦t ❤❛✈❡ ❛ ❍❛♠✐❧t♦♥✐❛♥ ❝②❝❧❡✱ ❛♥❞ ❤❛✈❡ ❛ ❍❛♠✐❧t♦♥✐❛♥ ♣❛t❤✿
✇✐t❤ ❛ tr✐❛♥❣❧❡ ❛♥❞ ❥♦✐♥✐♥❣ t❤❡ ✈❡rt✐❝❡s ✐♥ ❛ ♥❛t✉r❛❧ ✇❛②
(1970)✿ st✐❧❧ ♦♣❡♥
❊✈❡r② ❝♦♥♥❡❝t❡❞ ❈❛②❧❡② ❣r❛♣❤ ♦♥ ❛ ✜♥✐t❡ ❣r♦✉♣ ❤❛s ❛ ❍❛♠✐❧t♦♥✐❛♥ ❝②❝❧❡✳
❊❧❡♥❛ ❑♦♥st❛♥t✐♥♦✈❛ Pr♦♣❡rt✐❡s ♦❢ t❤❡ ❙t❛r ❣r❛♣❤s ❙❏❚❯✲✷✵✶✽ ✸✹ ✴ ✸✺
❊❧❡♥❛ ❑♦♥st❛♥t✐♥♦✈❛ Pr♦♣❡rt✐❡s ♦❢ t❤❡ ❙t❛r ❣r❛♣❤s ❙❏❚❯✲✷✵✶✽ ✸✺ ✴ ✸✺