P❛tt❡r♥ ❛✈♦✐❞❛♥❝❡ ✐♥ ♣❛rt✐❛❧ ♣❡r♠✉t❛t✐♦♥s ❆♥❞❡rs ❈❧❛❡ss♦♥ ❱✓ ✏t ❏❡❧✓ ✏♥❡❦ ❊✈❛ ❏❡❧✓ ✏♥❦♦✈✓ ❛ ❙❡r❣❡② ❑✐t❛❡✈ ❘❡②❦❥❛✈✐❦ ❯♥✐✈❡rs✐t②
❆❜str❛❝t ▼♦t✐✈❛t❡❞ ❜② t❤❡ ❝♦♥❝❡♣t ♦❢ ♣❛rt✐❛❧ ✇♦r❞s✱ ✇❡ ✐♥tr♦❞✉❝❡ ❛♥ ❛♥❛❧♦❣♦✉s ❝♦♥❝❡♣t ♦❢ ♣❛rt✐❛❧ ♣❡r♠✉t❛t✐♦♥s✳ ❲❡ ✐♥tr♦❞✉❝❡ ♣❛tt❡r♥✲❛✈♦✐❞❛♥❝❡ ✐♥ ♣❛rt✐❛❧ ♣❡r♠✉t❛t✐♦♥s ❛♥❞ ♣r♦✈❡ t❤❛t ♠♦st ❦♥♦✇♥ r❡s✉❧ts ♦♥ ❲✐❧❢ ❡q✉✐✈❛❧❡♥❝❡ ♦❢ ♣❡r♠✉t❛t✐♦♥ ♣❛tt❡r♥s ❝❛♥ ❜❡ ❡①t❡♥❞❡❞ t♦ ♣❛rt✐❛❧ ♣❡r♠✉t❛t✐♦♥s ✇✐t❤ ❛♥ ❛r❜✐tr❛r② ♥✉♠❜❡r ♦❢ ❤♦❧❡s✳ ❲❡ ❛❧s♦ s❤♦✇ t❤❛t ❇❛①t❡r ♣❡r♠✉t❛t✐♦♥s ♦❢ ❛ ❣✐✈❡♥ ❧❡♥❣t❤ ❦ ❝♦rr❡s♣♦♥❞ t♦ ❛ ❲✐❧❢✲t②♣❡ ❡q✉✐✈❛❧❡♥❝❡ ❝❧❛ss ✇✐t❤ r❡s♣❡❝t t♦ ♣❛rt✐❛❧ ♣❡r♠✉t❛t✐♦♥s ✇✐t❤ ✭ ❦ � ✷✮ ❤♦❧❡s✳ ▲❛st❧②✱ ✇❡ ❡♥✉♠❡r❛t❡ t❤❡ ♣❛rt✐❛❧ ♣❡r♠✉t❛t✐♦♥s ♦❢ ❧❡♥❣t❤ ♥ ✇✐t❤ ❦ ❤♦❧❡s ❛✈♦✐❞✐♥❣ ❛ ❣✐✈❡♥ ♣❛tt❡r♥ ♦❢ ❧❡♥❣t❤ ❛t ♠♦st ✹✱ ❢♦r ❡❛❝❤ ♥ ✕ ❦ ✕ ✶✳
P❛rt✐❛❧ ♣❡r♠✉t❛t✐♦♥s ❆ partial permutation of length ♥ with ❦ holes ✐s ❛ ✇♦r❞ ✙ ❂ ❛ ✶ ❛ ✷ ✿ ✿ ✿ ❛ ♥ ✐♥ ✇❤✐❝❤ ❡❛❝❤ ♦❢ t❤❡ ❡❧❡♠❡♥ts ❬ ♥ � ❦ ❪ ❂ ❢ ✶ ❀ ✷ ❀ ✿ ✿ ✿ ❀ ♥ � ❦ ❣ ❛♣♣❡❛rs ❡①❛❝t❧② ♦♥❝❡❀ t❤❡ r❡♠❛✐♥✐♥❣ ❦ s②♠❜♦❧s ♦❢ ✙ ❛r❡ ❭❤♦❧❡s✧✳ ❋♦r ❡①❛♠♣❧❡✱ ✸ ✆✆ ✹ ✆ ✶✷ ✐s ❛ ❧❡♥❣t❤ ✼ ♣❛rt✐❛❧ ♣❡r♠✉t❛t✐♦♥ ♦❢ ❢ ✶ ❀ ✷ ❀ ✸ ❀ ✹ ❣ ✇✐t❤ ✸ ❤♦❧❡s✳
▲❡t ❙ ❦ ♥ ❂ ♣❛rt✐❛❧ ♣❡r♠✉t❛t✐♦♥s ♦❢ ❬ ♥ � ❦ ❪ ✇✐t❤ ❡①❛❝t❧② ❦ ❤♦❧❡s✳ ❙ ✵ ✸ ❂ ❢ ✶✷✸ ❀ ✶✸✷ ❀ ✷✶✸ ❀ ✷✸✶ ❀ ✸✶✷ ❀ ✸✷✶ ❣ ❙ ✶ ✸ ❂ ❢ ✶✷ ✆ ❀ ✶ ✆ ✷ ❀ ✷✶ ✆ ❀ ✷ ✆ ✶ ❀ ✆ ✶✷ ❀ ✆ ✷✶ ❣ ❙ ✷ ✸ ❂ ❢ ✶ ✆✆ ❀ ✆ ✶ ✆ ❀ ✆✆ ✶ ❣ ❙ ✸ ✸ ❂ ❢✆✆✆❣ ◆♦t❡ t❤❛t ✥ ✦ ♥ ❥❙ ❦ ♥ ❥ ❂ ✭ ♥ � ❦ ✮✦ ❂ ♥ ✦ ❂❦ ✦ ❦ ◆♦t❡ ❛❧s♦ t❤❛t ❙ ✵ ♥ ✐s t❤❡ ❢❛♠✐❧✐❛r s②♠♠❡tr✐❝ ❣r♦✉♣ ❙ ♥ ✳ ❋♦r ❍ ✒ ❬ ♥ ❪✱ ❧❡t ♥ ❂ ❢ ❛ ✶ ✿ ✿ ✿ ❛ ♥ ✷ ❙ ❥ ❍ ❥ ❙ ❍ ✿ ❛ ✐ ❂ ✆ ✭ ✮ ✐ ✷ ❍ ❣ ✿ ♥
P❛tt❡r♥ ❛✈♦✐❞❛♥❝❡ ✐♥ ♣❛rt✐❛❧ ♣❡r♠✉t❛t✐♦♥s ▲❡t ✐ ✶ ❁ ✁ ✁ ✁ ❁ ✐ ♥ � ❦ ❜❡ t❤❡ ✐♥❞✐❝❡s ♦❢ t❤❡ ♥♦♥✲❤♦❧❡ ❡❧❡♠❡♥ts ♦❢ ✙ ✷ ❙ ❦ ♥ ✳ ❆ ♣❡r♠✉t❛t✐♦♥ ✛ ✷ ❙ ♥ ✐s ❛♥ extension ♦❢ ✙ ✐❢ ✛ ✭ ✐ ✶ ✮ ✿ ✿ ✿ ✛ ✭ ✐ ♥ � ❦ ✮ ✐s ♦r❞❡r ✐s♦♠♦r♣❤✐❝ t♦ ✙ ✭ ✐ ✶ ✮ ✿ ✿ ✿ ✙ ✭ ✐ ♥ � ❦ ✮ ✿ ❊✳❣✳✱ ✷ ✆ ✶ ❤❛s t❤r❡❡ ❡①t❡♥s✐♦♥s✿ ✸✶✷✱ ✸✷✶ ❛♥❞ ✷✸✶✳ ✙ ✷ ❙ ❦ ♥ avoids ♣ ✷ ❙ ❵ ✐❢ ❡❛❝❤ ❡①t❡♥s✐♦♥ ♦❢ ✙ ❛✈♦✐❞s ♣ ❊✳❣✳✱ ✙ ❂ ✸✷ ✆ ✶✺✹ ❛✈♦✐❞s ✶✷✸✹✱ ❜✉t ✙ ❞♦❡s ♥♦t ❛✈♦✐❞ ✶✷✸✿ t❤❡ ♣❡r♠✉t❛t✐♦♥ ✸✷✺✶✻✹ ✐s ❛♥ ❡①t❡♥s✐♦♥ ♦❢ ✙ ❛♥❞ ✐t ❝♦♥t❛✐♥s t✇♦ ♦❝❝✉rr❡♥❝❡s ♦❢ ✶✷✸✳ ❙ ❦ ♥ ✭ ♣ ✮ ❂ s❡t ♦❢ ♣ ✲❛✈♦✐❞✐♥❣ ♣❛rt✐❛❧ ♣❡r♠✉t❛t✐♦♥s ✐♥ ❙ ❦ ♥ ❙ ❍ ♥ ✭ ♣ ✮ ❂ s❡t ♦❢ ♣ ✲❛✈♦✐❞✐♥❣ ♣❛rt✐❛❧ ♣❡r♠✉t❛t✐♦♥s ✐♥ ❙ ❍ ♥
❦ ✲❲✐❧❢✲❡q✉✐✈❛❧❡♥❝❡ ❛♥❞ ❄ ✲❲✐❧❢✲❡q✉✐✈❛❧❡♥❝❡ ❚✇♦ ♣❛tt❡r♥s ♣ ❛♥❞ q ❛r❡ ◮ ❦ -Wilf-equivalent ✐❢ ❥❙ ❦ ♥ ✭ ♣ ✮ ❥ ❂ ❥❙ ❦ ♥ ✭ q ✮ ❥ ❢♦r ❛❧❧ ♥ ✳ ✭✵✲❲✐❧❢✲❡q✉✐✈❛❧❡♥❝❡ ✐s st❛♥❞❛r❞ ❲✐❧❢✲❡q✉✐✈❛❧❡♥❝❡✮ ◮ ❄ -Wilf-equivalent ✐❢ ♣ ❛♥❞ q ❛r❡ ❦ ✲❲✐❧❢✲❡q✉✐✈❛❧❡♥t ❢♦r ❛❧❧ ❦ ✳ ◮ strongly ❦ -Wilf-equivalent ✐❢ ❥❙ ❍ ♥ ✭ ♣ ✮ ❥ ❂ ❥❙ ❍ ♥ ✭ q ✮ ❥ ❢♦r ❡❛❝❤ ♥ ❛♥❞ ❢♦r ❡❛❝❤ ❦ ✲❡❧❡♠❡♥t s✉❜s❡t ❍ ✒ ❬ ♥ ❪✳ ◮ strongly ❄ -Wilf-equivalent ✐❢ t❤❡② ❛r❡ str♦♥❣❧② ❦ ✲❲✐❧❢✲❡q✉✐✈❛❧❡♥t ❢♦r ❛❧❧ ❦ ✳
◆♦t❡ t❤❛t ◮ str♦♥❣ ❦ ✲❲✐❧❢✲❡q✉✐✈❛❧❡♥❝❡ ❂ ✮ ❦ ✲❲✐❧❢✲❡q✉✐✈❛❧❡♥❝❡ ◮ str♦♥❣ ❄ ✲❲✐❧❢✲❡q✉✐✈❛❧❡♥❝❡ ❂ ✮ ❄ ✲❲✐❧❢✲❡q✉✐✈❛❧❡♥❝❡ ❚❤❡ ❝♦♥✈❡rs❡ ✐♠♣❧✐❝❛t✐♦♥s ❛r❡ ♥♦t tr✉❡✳ ❈♦♥s✐❞❡r ♣ ❂ ✶✸✹✷ ❛♥❞ q ❂ ✷✹✸✶✳ ❆ ♣❛rt✐❛❧ ♣❡r♠✉t❛t✐♦♥ ❛✈♦✐❞s ♣ ✐☛ ✐ts r❡✈❡rs❡ ❛✈♦✐❞s q ✱ ❛♥❞ t❤✉s ♣ ❛♥❞ q ❛r❡ ❄ ✲❲✐❧❢✲❡q✉✐✈❛❧❡♥t✳ ❍♦✇❡✈❡r✱ ♣ ❛♥❞ q ❛r❡ ♥♦t str♦♥❣❧② ✶✲❲✐❧❢✲❡q✉✐✈❛❧❡♥t✱ ❛♥❞ ❤❡♥❝❡ ♥♦t str♦♥❣❧② ❄ ✲❲✐❧❢✲❡q✉✐✈❛❧❡♥t ❡✐t❤❡r✳ ❚♦ s❡❡ t❤✐s✿ ❧❡t ❍ ❂ ❢ ✷ ❣ ❀ ❝❤❡❝❦ ❥❙ ❍ ✺ ✭ ♣ ✮ ❥ ❂ ✶✸ ✻ ❂ ❥❙ ❍ ✺ ✭ q ✮ ❥ ❂ ✶✹✳
❲❡ s❤♦✇ t❤❛t ♠♦st ♣❛✐rs ♦❢ ❲✐❧❢✲❡q✉✐✈❛❧❡♥t ♣❛tt❡r♥s t❤❛t ✇❡r❡ ❞✐s❝♦✈❡r❡❞ s♦ ❢❛r ❛r❡ ✐♥ ❢❛❝t ❄ ✲❲✐❧❢✲❡q✉✐✈❛❧❡♥t✳ ❚❤❡ ♦♥❧② ❡①❝❡♣t✐♦♥ ✐s t❤❡ ♣❛✐r ♣ ❂ ✷✹✶✸ ❛♥❞ q ❂ ✶✸✹✷✳ ❚❤❡s❡ ♣❛tt❡r♥s ❛r❡ ❲✐❧❢✲❡q✉✐✈❛❧❡♥t ❬✹❪✱ ❜✉t t❤❡② ❛r❡ ♥❡✐t❤❡r ✶✲❲✐❧❢✲❡q✉✐✈❛❧❡♥t ♥♦r ✷✲❲✐❧❢✲❡q✉✐✈❛❧❡♥t✳
❙tr♦♥❣ ❄ ✲❲✐❧❢✲❡q✉✐✈❛❧❡♥❝❡ ♦❢ ✶ ✷ ✿ ✿ ✿ ❵ ✛ ❛♥❞ ❵ ✿ ✿ ✿ ✷ ✶ ✛ ❚❤❡♦r❡♠ ❋♦r ❛♥② ❵ ✔ ♠ ✱ ❛♥❞ ❛♥② ♣❡r♠✉t❛t✐♦♥ ✛ ♦❢ ❢ ❵ ✰ ✶ ❀ ❵ ✰ ✷ ❀ ✿ ✿ ✿ ❀ ♠ ❣ ✱ ✶ ✷ ✿ ✿ ✿ ✭ ❵ � ✶✮ ❵ ✛ ✐s str♦♥❣❧② ❄ ✲❲✐❧❢✲❡q✉✐✈❛❧❡♥t t♦ ❵ ✭ ❵ � ✶✮ ✁ ✁ ✁ ✷ ✶ ✛ ✳ ❚❤✐s ✐s ❛ str❡♥❣t❤❡♥✐♥❣ ♦❢ ❛ r❡s✉❧t ♦❢ ❇❛❝❦❡❧✐♥✱ ❲❡st ❛♥❞ ❳✐♥ ❬✶❪✱ ✇❤♦ s❤♦✇ t❤❛t ♣❛tt❡r♥s ♦❢ t❤✐s ❢♦r♠ ❛r❡ ❲✐❧❢✲❡q✉✐✈❛❧❡♥t✳ ❖✉r ♣r♦♦❢ ✐s ❜❛s❡❞ ♦♥ ❛ ❞✐☛❡r❡♥t ❛r❣✉♠❡♥t t❤❛♥ t❤❡✐rs✳ ❚❤❡ ♠❛✐♥ ✐♥❣r❡❞✐❡♥t ♦❢ ♦✉r ♣r♦♦❢ ✐s ❛♥ ✐♥✈♦❧✉t✐♦♥ ♦♥ ❛ s❡t ♦❢ ☞❧❧✐♥❣s ♦❢ ❋❡rr❡rs ❞✐❛❣r❛♠s✱ ❞✐s❝♦✈❡r❡❞ ❜② ❑r❛tt❡♥t❤❛❧❡r ❬✸❪✳
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