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  1. ◆✉♠❜❡r ♦❢ ❢r❡❡ ✈❛r✐❛❜❧❡s ❙♣❡❝✐✜❝❛t✐♦♥s ✇✐t❤ ✈❛r✐♦✉s ♣❛r❛♠❡t❡rs ❙♦♠❡ ♣r♦❜❧❡♠s t♦ ❛❞❞r❡ss ❘❡❢❡r❡♥❝❡s ❙♦♠❡ ♣❛r❛♠❡t❡rs ✐♥ ❧✐♥❡❛r ❧❛♠❜❞❛✲t❡r♠s ❏❡❛♥ P❡②❡♥ ❏♦✐♥t ✇♦r❦ ✇✐t❤ ❖❧✐✈✐❡r ❇♦❞✐♥✐ ❛♥❞ ❙❡r❣❡② ❉♦✈❣❛❧ ❈▲❆ ✷✵✶✽✱ ❙♦r❜♦♥♥❡ ❯♥✐✈❡rs✐té

  2. ◆✉♠❜❡r ♦❢ ❢r❡❡ ✈❛r✐❛❜❧❡s ❙♣❡❝✐✜❝❛t✐♦♥s ✇✐t❤ ✈❛r✐♦✉s ♣❛r❛♠❡t❡rs ❙♦♠❡ ♣r♦❜❧❡♠s t♦ ❛❞❞r❡ss ❘❡❢❡r❡♥❝❡s ❚❛❜❧❡ ♦❢ ❝♦♥t❡♥ts ◆✉♠❜❡r ♦❢ ❢r❡❡ ✈❛r✐❛❜❧❡s ✶ ❙♣❡❝✐✜❝❛t✐♦♥ ❈♦rr❡s♣♦♥❞❡♥❝❡ ❜❡t✇❡❡♥ ♠❛♣s ❛♥❞ ❧✐♥❡❛r ❧❛♠❜❞❛✲t❡r♠s ❆s②♠♣t♦t✐❝ ❞✐str✐❜✉t✐♦♥ ❙♣❡❝✐✜❝❛t✐♦♥s ✇✐t❤ ✈❛r✐♦✉s ♣❛r❛♠❡t❡rs ✷ ❙♣❡❝✐✜❝❛t✐♦♥s ❆s②♠♣t♦t✐❝ ❞✐str✐❜✉t✐♦♥ ❙♦♠❡ ♣r♦❜❧❡♠s t♦ ❛❞❞r❡ss ✸ ❘❡❢❡r❡♥❝❡s ✹

  3. ◆✉♠❜❡r ♦❢ ❢r❡❡ ✈❛r✐❛❜❧❡s ❙♣❡❝✐✜❝❛t✐♦♥s ✇✐t❤ ✈❛r✐♦✉s ♣❛r❛♠❡t❡rs ❙♦♠❡ ♣r♦❜❧❡♠s t♦ ❛❞❞r❡ss ❘❡❢❡r❡♥❝❡s ❈♦✉♥t✐♥❣ ♣❧❛✐♥ ❛♥❞ ❝❧♦s❡❞ t❡r♠s ❆ ♣❧❛✐♥ ❧✐♥❡❛r✲❧❛♠❜❞❛ t❡r♠ ❝❛♥ ❡✐t❤❡r ❜❡ ❛♥ ❛♣♣❧✐❝❛t✐♦♥ ❜❡t✇❡❡♥ t✇♦ t❡r♠s✱ ♦r ❛ s♠❛❧❧❡r t❡r♠ ✇❤❡r❡ ❛ ❜✐♥❛r② ♥♦❞❡ ✐s ✐♥s❡rt❡❞ ♦♥ t❤❡ t♦♣ ❧❡❢t ♦r ♦♥ t❤❡ t♦♣ r✐❣❤t ♦❢ ❛♥ ❡①✐st✐♥❣ ♥♦❞❡✱ ❛♥ ❛❜str❛❝t✐♦♥ ✐s ❛❞❞❡❞ ❛❜♦✈❡ t❤❡ r♦♦t ❛♥❞ ❝♦♥♥❡❝t❡❞ t♦ t❤❡ ♥❡✇ ❧❡❛❢✿ • • � T n +3 = p,l T ( f ) T ( f ) InsertBin T n p n +2 − p f ( z ) = z + z 2 + zf 2 ( z ) + 2 z 4 ∂ z f ( z ) . ❈❧♦s❡❞ t❡r♠s ♦❜❡② t❤❡ s❛♠❡ s♣❡❝✐✜❝❛t✐♦♥ ❛♣❛rt ❢r♦♠ t❤❡ ✐♥✐t✐❛❧ t❡r♠s✿ f ( z ) = z 2 + zf 2 ( z ) + 2 z 4 ∂ z f ( z ) .

  4. ◆✉♠❜❡r ♦❢ ❢r❡❡ ✈❛r✐❛❜❧❡s ❙♣❡❝✐✜❝❛t✐♦♥s ✇✐t❤ ✈❛r✐♦✉s ♣❛r❛♠❡t❡rs ❙♦♠❡ ♣r♦❜❧❡♠s t♦ ❛❞❞r❡ss ❘❡❢❡r❡♥❝❡s ❈♦♥str✉❝t✐♦♥ ♦❢ ❝❧♦s❡❞ t❡r♠s ♦❢ s✐③❡ ✺ ❢r♦♠ t❤❡ t❡r♠ ♦❢ s✐③❡ ✷✳ • • • • • • • • • • • • • • • • • • • • • • • • • • •

  5. ◆✉♠❜❡r ♦❢ ❢r❡❡ ✈❛r✐❛❜❧❡s ❙♣❡❝✐✜❝❛t✐♦♥s ✇✐t❤ ✈❛r✐♦✉s ♣❛r❛♠❡t❡rs ❙♦♠❡ ♣r♦❜❧❡♠s t♦ ❛❞❞r❡ss ❘❡❢❡r❡♥❝❡s ❲❡ ❝❛♥ ❛❧s♦ ❣✐✈❡ ❛ s♣❡❝✐✜❝❛t✐♦♥ ✇❤❡r❡ ✇❡ t❛❦❡ ✐♥t♦ ❛❝❝♦✉♥t t❤❡ ♥✉♠❜❡r ♦❢ ❢r❡❡ ✈❛r✐❛❜❧❡s✿ • • � T ( f ) n +1 ,m = p,l T ( f ) T ( f ) T ( f ) n,m +1 n − p,m − l p,l f ( z, u ) = uz + zf 2 ( z, u ) + 2 ∂ u f ( z, u )

  6. ◆✉♠❜❡r ♦❢ ❢r❡❡ ✈❛r✐❛❜❧❡s ❙♣❡❝✐✜❝❛t✐♦♥s ✇✐t❤ ✈❛r✐♦✉s ♣❛r❛♠❡t❡rs ❙♦♠❡ ♣r♦❜❧❡♠s t♦ ❛❞❞r❡ss ❘❡❢❡r❡♥❝❡s ❙♦♠❡ ❝♦❡✣❝✐❡♥ts ♦❢ t❤❡ ❡♥✉♠❡r❛t✐♥❣ ❛rr❛②✳ ❍❍❍❍ ♠ ✵ ✶ ✷ ✸ ✹ ♥ ❍ ✶ ✵ ✶ ✵ ✵ ✵ ✷ ✶ ✵ ✵ ✵ ✵ ✸ ✵ ✵ ✶ ✵ ✵ ✹ ✵ ✹ ✵ ✵ ✵ ✺ ✺ ✵ ✵ ✷ ✵ ✻ ✵ ✵ ✶✻ ✵ ✵ ✼ ✵ ✺✵ ✵ ✵ ✺ ✽ ✻✵ ✵ ✵ ✻✹ ✵ ✾ ✵ ✵ ✸✺✵ ✵ ✵ ✶✵ ✵ ✾✻✵ ✵ ✵ ✷✺✻ ✶✶ ✶✶✵✺ ✵ ✵ ✷✶✵✵ ✵

  7. ◆✉♠❜❡r ♦❢ ❢r❡❡ ✈❛r✐❛❜❧❡s ❙♣❡❝✐✜❝❛t✐♦♥s ✇✐t❤ ✈❛r✐♦✉s ♣❛r❛♠❡t❡rs ❙♦♠❡ ♣r♦❜❧❡♠s t♦ ❛❞❞r❡ss ❘❡❢❡r❡♥❝❡s ❈♦rr❡s♣♦♥❞❡♥❝❡ ✇✐t❤ tr✐✈❛❧❡♥t ♠❛♣s ❉❡✜♥✐t✐♦♥ ❯♣ t♦ ❛ ❧❛❜❡❧❧✐♥❣✱ ❛ r♦♦t❡❞ tr✐✈❛❧❡♥t ♠❛♣ ♦❢ s✐③❡ n = 6 k + 2 ✐s ❛ r♦♦t ✐♥ ❏ 1 , n ❑ ❛♥❞ ❝♦✉♣❧❡ ♦❢ t✇♦ ♣❡r♠✉t❛t✐♦♥s ♦✈❡r ❏ 1 , n ❑ ✱ s✉❝❤ t❤❛t t❤❡ ✜rst ♦♥❡✱ ❞❡♥♦t❡❞ ❜② σ • ✱ ✐s ❛♥ ✐♥✈♦❧✉t✐♦♥ ✇✐t❤ ♥♦ ✜①❡❞ ♣♦✐♥t✱ t❤❡ s❡❝♦♥❞ ♦♥❡✱ ❞❡♥♦t❡❞ ❜② σ • ✱ ✐s ♦❢ ♦r❞❡r 3 ❛♥❞ ❤❛s t✇♦ ✜①❡❞ ♣♦✐♥ts a ❛♥❞ b ✭✐♥❝❧✉❞✐♥❣ t❤❡ r♦♦t✮✱ ❛♥❞ s✉❝❤ t❤❛t σ • ✱ σ • ❛♥❞ ( a b ) ❣❡♥❡r❛t❡ ❛ tr❛♥s✐t✐✈❡ ❣r♦✉♣✳

  8. ◆✉♠❜❡r ♦❢ ❢r❡❡ ✈❛r✐❛❜❧❡s ❙♣❡❝✐✜❝❛t✐♦♥s ✇✐t❤ ✈❛r✐♦✉s ♣❛r❛♠❡t❡rs ❙♦♠❡ ♣r♦❜❧❡♠s t♦ ❛❞❞r❡ss ❘❡❢❡r❡♥❝❡s ❘❡♠❛r❦ ❲❡ ❝❛♥ r❡♣r❡s❡♥t r♦♦t❡❞ tr✐✈❛❧❡♥t ♠❛♣s ✇✐t❤ ❣r❛♣❤s ✇✐t❤ ❞❡❣r❡❡ 2 ♥♦❞❡s • ❛♥❞ ❞❡❣r❡❡ 3 ♥♦❞❡s • ✇❤❡r❡ σ • ✱ σ • ❛❝t ♦♥ t❤❡ ❡❞❣❡s✳ ❊①❛♠♣❧❡ ρ = 7 ✱ σ • = (12)(38)(56)(74) ✱ σ • = (123)(456) ✳ ✺ ✶ 4 7 ✽ ✸ • • • • • • • ✻ ✷

  9. ◆✉♠❜❡r ♦❢ ❢r❡❡ ✈❛r✐❛❜❧❡s ❙♣❡❝✐✜❝❛t✐♦♥s ✇✐t❤ ✈❛r✐♦✉s ♣❛r❛♠❡t❡rs ❙♦♠❡ ♣r♦❜❧❡♠s t♦ ❛❞❞r❡ss ❘❡❢❡r❡♥❝❡s ❚❤❡♦r❡♠ ✭❖✳❇♦❞✐♥✐✱ ❉✳ ●❛r❞②✱ ❆✳ ❏❛❝q✉♦t✱ ✷✵✶✸✮ T ( f ) 3 k +2 , 0 ✐s ✐♥ ❜✐❥❡❝t✐♦♥ ✇✐t❤ tr✐✈❛❧❡♥t r♦♦t❡❞ ♠❛♣✳ ■♥ ♣❛rt✐❝✉❧❛r t❤❡ ❣❡♥❡r✲ ❛t✐♥❣ ❢✉♥❝t✐♦♥ ♦❢ ❝❧♦s❡❞ t❡r♠s ✐s ❣✐✈❡♥ ❜②✿ f = [ z 6 k +2 ] z 3 ∂ z log( e z 3 / 3 ⊙ e z 2 / 2 ) [ z 3 k +2 , u 0 ] � ✇❤❡r❡ ⊙ ✐s t❤❡ ❡①♣♦♥❡♥t✐❛❧ ❍❛❞❛♠❛r❞ ♣r♦❞✉❝t ✐♥ z ✳

  10. ◆✉♠❜❡r ♦❢ ❢r❡❡ ✈❛r✐❛❜❧❡s ❙♣❡❝✐✜❝❛t✐♦♥s ✇✐t❤ ✈❛r✐♦✉s ♣❛r❛♠❡t❡rs ❙♦♠❡ ♣r♦❜❧❡♠s t♦ ❛❞❞r❡ss ❘❡❢❡r❡♥❝❡s ❚❤❡♦r❡♠ ✭❈♦♥s❡q✉❡♥❝❡ ♦❢ t❤❡ ❝♦rr❡s♣♦♥❞❡♥❝❡ ♦❢ ◆✳ ❩❡✐❧❜❡r❣❡r✱ ✷✵✶✼✮ ❲❡ ❝❛♥ ❣❡♥❡r❛❧✐③❡ t❤✐s ❝♦rr❡s♣♦♥❞❡♥❝❡ t♦ ♣❧❛✐♥ t❡r♠s✳ ■♥ ♣❛rt✐❝✉❧❛r✱ t❤❡ ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥ ♦❢ T ( f ) ✐s ❣✐✈❡♥ ❜②✿ c ( m ) , u m ] z 3 ∂ z log( e z 3 / 3+ uz ⊙ e z 2 / 2 ) [ z 3 k + c ( m ) , u m ] � f = [ z 6 k + � ✇❤❡r❡ c ( m ) = 2 ✐❢ m = 0 mod [3] ✱ 1 ✐❢ m = 1 mod [3] ✱ 0 ✐❢ m = 2 mod [3] � c ( m ) = 2 ✐❢ m = 0 mod [3] ✱ 6 ✐❢ m = 1 mod [3] ✱ 4 ✐❢ m = 2 mod [3] ❘❡♠❛r❦ ❚❤❛♥❦s t♦ ❛ t❤❡♦r❡♠ ♦❢ ❊✳ ❆✳ ❇❡♥❞❡r✱ ✇❡ ❝❛♥ ♣r♦✈❡ t❤❡ s✉❜s❡q✉❡♥t ♣r♦♣♦s✐t✐♦♥ ✉s✐♥❣ t❤❡ ❝♦❡✣❝✐❡♥ts ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ❢✉♥❝t✐♦♥✿ e z 3 / 3+ uz ⊙ e z 2 / 2

  11. ◆✉♠❜❡r ♦❢ ❢r❡❡ ✈❛r✐❛❜❧❡s ❙♣❡❝✐✜❝❛t✐♦♥s ✇✐t❤ ✈❛r✐♦✉s ♣❛r❛♠❡t❡rs ❙♦♠❡ ♣r♦❜❧❡♠s t♦ ❛❞❞r❡ss ❘❡❢❡r❡♥❝❡s ❈♦rr❡s♣♦♥❞❡♥❝❡ ❜❡t✇❡❡♥ ♠❛♣s ❛♥❞ ❝❧♦s❡❞ t❡r♠s • ■♥✐t✐❛❧ t❡r♠✿ • • ρ ≃ • • • ❆♣♣❧✐❝❛t✐♦♥ ✭❋❧✐♣ ❝❤❛♥❣❡s r♦♦t ♥♦❞❡ ✐♥t♦ ❛ ❜❧❛❝❦ ♥♦❞❡ ✇✐t❤ ♥♦ r♦♦t ❡❞❣❡✮✿ • ρ • • • ≃ ❋❧✐♣ ❆ ❋❧✐♣ ❇ ❆ ❇ • ❆❞❞ ❛♥ ❛❜str❛❝t✐♦♥ ❛❜♦✈❡ t❤❡ r♦♦t ✭✐❢ ❆ ✐s ♥♦t ❝❧♦s❡❞✮✿ • • ρ • ≃ ❆ ❋❧✐♣ ❆

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