PermPal Database

Av(1234, 1243, 1432)

Generating Function

- \frac{x^{6} + 5 x^{4} - 12 x^{3} + 12 x^{2} - 6 x + 1}{2 x^{5} - 13 x^{4} + 21 x^{3} - 17 x^{2} + 7 x - 1}

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Counting Sequence

1, 1, 2, 6, 21, 76, 278, 1021, 3756, 13827, 50916, 187512, 690593, 2543444, 9367525, ...

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Implicit Equation for the Generating Function

(2 x^{5} - 13 x^{4} + 21 x^{3} - 17 x^{2} + 7 x - 1) F (x) + x^{6} + 5 x^{4} - 12 x^{3} + 12 x^{2} - 6 x + 1 = 0

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Recurrence

a (0) = 1

a (1) = 1

a (2) = 2

a (3) = 6

a (4) = 21

a (5) = 76

a (6) = 278

a (n + 5) = 2 a (n) - 13 a (n + 1) + 21 a (n + 2) - 17 a (n + 3) + 7 a (n + 4), n \geq 7

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Explicit Closed Form

\frac{37364 RootOf {(2 Z^{5} - 13 Z^{4} + 21 Z^{3} - 17 Z^{2} + 7 Z - 1, index = 1)}^{- n}}{126571} + \frac{37364 RootOf {(2 Z^{5} - 13 Z^{4} + 21 Z^{3} - 17 Z^{2} + 7 Z - 1, index = 2)}^{- n}}{126571} + \frac{37364 RootOf {(2 Z^{5} - 13 Z^{4} + 21 Z^{3} - 17 Z^{2} + 7 Z - 1, index = 3)}^{- n}}{126571} + \frac{37364 RootOf {(2 Z^{5} - 13 Z^{4} + 21 Z^{3} - 17 Z^{2} + 7 Z - 1, index = 4)}^{- n}}{126571} + \frac{37364 RootOf {(2 Z^{5} - 13 Z^{4} + 21 Z^{3} - 17 Z^{2} + 7 Z - 1, index = 5)}^{- n}}{126571} - \frac{60644 RootOf {(2 Z^{5} - 13 Z^{4} + 21 Z^{3} - 17 Z^{2} + 7 Z - 1, index = 1)}^{1 - n}}{126571} - \frac{60644 RootOf {(2 Z^{5} - 13 Z^{4} + 21 Z^{3} - 17 Z^{2} + 7 Z - 1, index = 2)}^{1 - n}}{126571} - \frac{60644 RootOf {(2 Z^{5} - 13 Z^{4} + 21 Z^{3} - 17 Z^{2} + 7 Z - 1, index = 3)}^{1 - n}}{126571} - \frac{60644 RootOf {(2 Z^{5} - 13 Z^{4} + 21 Z^{3} - 17 Z^{2} + 7 Z - 1, index = 4)}^{1 - n}}{126571} - \frac{60644 RootOf {(2 Z^{5} - 13 Z^{4} + 21 Z^{3} - 17 Z^{2} + 7 Z - 1, index = 5)}^{1 - n}}{126571} - \frac{3041 RootOf {(2 Z^{5} - 13 Z^{4} + 21 Z^{3} - 17 Z^{2} + 7 Z - 1, index = 1)}^{- n - 1}}{126571} - \frac{3041 RootOf {(2 Z^{5} - 13 Z^{4} + 21 Z^{3} - 17 Z^{2} + 7 Z - 1, index = 2)}^{- n - 1}}{126571} - \frac{3041 RootOf {(2 Z^{5} - 13 Z^{4} + 21 Z^{3} - 17 Z^{2} + 7 Z - 1, index = 3)}^{- n - 1}}{126571} - \frac{3041 RootOf {(2 Z^{5} - 13 Z^{4} + 21 Z^{3} - 17 Z^{2} + 7 Z - 1, index = 4)}^{- n - 1}}{126571} - \frac{3041 RootOf {(2 Z^{5} - 13 Z^{4} + 21 Z^{3} - 17 Z^{2} + 7 Z - 1, index = 5)}^{- n - 1}}{126571} + \frac{60239 RootOf {(2 Z^{5} - 13 Z^{4} + 21 Z^{3} - 17 Z^{2} + 7 Z - 1, index = 1)}^{- n + 2}}{126571} + \frac{60239 RootOf {(2 Z^{5} - 13 Z^{4} + 21 Z^{3} - 17 Z^{2} + 7 Z - 1, index = 2)}^{- n + 2}}{126571} + \frac{60239 RootOf {(2 Z^{5} - 13 Z^{4} + 21 Z^{3} - 17 Z^{2} + 7 Z - 1, index = 3)}^{- n + 2}}{126571} + \frac{60239 RootOf {(2 Z^{5} - 13 Z^{4} + 21 Z^{3} - 17 Z^{2} + 7 Z - 1, index = 4)}^{- n + 2}}{126571} + \frac{60239 RootOf {(2 Z^{5} - 13 Z^{4} + 21 Z^{3} - 17 Z^{2} + 7 Z - 1, index = 5)}^{- n + 2}}{126571} - \frac{5384 RootOf {(2 Z^{5} - 13 Z^{4} + 21 Z^{3} - 17 Z^{2} + 7 Z - 1, index = 1)}^{- n + 3}}{126571} - \frac{5384 RootOf {(2 Z^{5} - 13 Z^{4} + 21 Z^{3} - 17 Z^{2} + 7 Z - 1, index = 2)}^{- n + 3}}{126571} - \frac{5384 RootOf {(2 Z^{5} - 13 Z^{4} + 21 Z^{3} - 17 Z^{2} + 7 Z - 1, index = 3)}^{- n + 3}}{126571} - \frac{5384 RootOf {(2 Z^{5} - 13 Z^{4} + 21 Z^{3} - 17 Z^{2} + 7 Z - 1, index = 4)}^{- n + 3}}{126571} - \frac{5384 RootOf {(2 Z^{5} - 13 Z^{4} + 21 Z^{3} - 17 Z^{2} + 7 Z - 1, index = 5)}^{- n + 3}}{126571} - \frac{({\begin{array}{cc} \frac{13}{2} & n = 0 \\ 1 & n = 1 \\ 0 & otherwise \end{array})}{2}

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This specification was found using the strategy pack "Point Placements" and has 183 rules.

Found on January 18, 2022.

Finding the specification took 26 seconds.

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\begin{aligned} F_{0} (x) & = F_{1} (x) + F_{2} (x) \\ F_{1} (x) & = 1 \\ F_{2} (x) & = F_{3} (x) \\ F_{3} (x) & = F_{4} (x) F_{5} (x) \\ F_{4} (x) & = x \\ F_{5} (x) & = F_{19} (x) + F_{6} (x) \\ F_{6} (x) & = F_{1} (x) + F_{7} (x) \\ F_{7} (x) & = F_{8} (x) \\ F_{8} (x) & = F_{4} (x) F_{9} (x) \\ F_{9} (x) & = F_{10} (x) + F_{11} (x) \\ F_{10} (x) & = F_{1} (x) + F_{4} (x) \\ F_{11} (x) & = F_{12} (x) + F_{14} (x) \\ F_{12} (x) & = F_{13} (x) \\ F_{13} (x) & = F_{10} (x) F_{4} (x) \\ F_{14} (x) & = F_{15} (x) + F_{16} (x) + F_{18} (x) \\ F_{15} (x) & = 0 \\ F_{16} (x) & = F_{17} (x) F_{4} (x) \\ F_{17} (x) & = F_{4} (x) \\ F_{18} (x) & = F_{12} (x) F_{4} (x) \\ F_{19} (x) & = F_{2} (x) + F_{20} (x) \\ F_{20} (x) & = F_{15} (x) + F_{181} (x) + F_{21} (x) \\ F_{21} (x) & = F_{22} (x) F_{4} (x) \\ F_{22} (x) & = F_{23} (x) + F_{34} (x) \\ F_{23} (x) & = F_{24} (x) + F_{7} (x) \\ F_{24} (x) & = F_{15} (x) + F_{25} (x) + F_{28} (x) \\ F_{25} (x) & = F_{26} (x) F_{4} (x) \\ F_{26} (x) & = F_{17} (x) + F_{27} (x) \\ F_{27} (x) & = F_{14} (x) \\ F_{28} (x) & = F_{29} (x) F_{4} (x) \\ F_{29} (x) & = F_{11} (x) + F_{30} (x) \\ F_{30} (x) & = F_{31} (x) + F_{32} (x) \\ F_{31} (x) & = F_{16} (x) \\ F_{32} (x) & = F_{33} (x) \\ F_{33} (x) & = F_{31} (x) F_{4} (x) \\ F_{34} (x) & = F_{20} (x) + F_{35} (x) \\ F_{35} (x) & = F_{140} (x) + F_{15} (x) + F_{179} (x) + F_{36} (x) \\ F_{36} (x) & = F_{37} (x) F_{4} (x) \\ F_{37} (x) & = F_{38} (x) + F_{46} (x) \\ F_{38} (x) & = F_{24} (x) + F_{39} (x) \\ F_{39} (x) & = F_{15} (x) + F_{40} (x) + F_{41} (x) + F_{44} (x) \\ F_{40} (x) & = 0 \\ F_{41} (x) & = F_{4} (x) F_{42} (x) \\ F_{42} (x) & = F_{27} (x) + F_{43} (x) \\ F_{43} (x) & = F_{32} (x) \\ F_{44} (x) & = F_{4} (x) F_{45} (x) \\ F_{45} (x) & = F_{30} (x) \\ F_{46} (x) & = F_{35} (x) + F_{47} (x) \\ F_{47} (x) & = F_{138} (x) + F_{15} (x) + F_{177} (x) + F_{48} (x) + F_{92} (x) \\ F_{48} (x) & = F_{4} (x) F_{49} (x) \\ F_{49} (x) & = F_{50} (x) + F_{54} (x) \\ F_{50} (x) & = F_{39} (x) + F_{51} (x) \\ F_{51} (x) & = F_{52} (x) \\ F_{52} (x) & = F_{4} (x) F_{53} (x) \\ F_{53} (x) & = F_{43} (x) \\ F_{54} (x) & = F_{47} (x) + F_{55} (x) \\ F_{55} (x) & = F_{15} (x) + F_{176} (x) + F_{56} (x) + F_{60} (x) + F_{61} (x) + F_{62} (x) \\ F_{56} (x) & = F_{4} (x) F_{57} (x) \\ F_{57} (x) & = F_{58} (x) + F_{59} (x) \\ F_{58} (x) & = F_{51} (x) \\ F_{59} (x) & = F_{55} (x) \\ F_{60} (x) & = 0 \\ F_{61} (x) & = 0 \\ F_{62} (x) & = F_{4} (x) F_{63} (x) \\ F_{63} (x) & = F_{64} (x) \\ F_{64} (x) & = F_{65} (x) \\ F_{65} (x) & = 2 F_{15} (x) + F_{66} (x) + F_{70} (x) + F_{71} (x) \\ F_{66} (x) & = F_{4} (x) F_{67} (x) \\ F_{67} (x) & = F_{68} (x) + F_{69} (x) \\ F_{68} (x) & = F_{32} (x) \\ F_{69} (x) & = F_{65} (x) \\ F_{70} (x) & = 0 \\ F_{71} (x) & = F_{4} (x) F_{72} (x) \\ F_{72} (x) & = 2 F_{15} (x) + F_{113} (x) + F_{73} (x) \\ F_{73} (x) & = F_{4} (x) F_{74} (x) \\ F_{74} (x) & = F_{75} (x) + F_{78} (x) \\ F_{75} (x) & = F_{31} (x) + F_{76} (x) \\ F_{76} (x) & = F_{77} (x) \\ F_{77} (x) & = F_{27} (x) F_{4} (x) \\ F_{78} (x) & = F_{72} (x) + F_{79} (x) \\ F_{79} (x) & = 2 F_{15} (x) + F_{80} (x) + F_{92} (x) + F_{93} (x) \\ F_{80} (x) & = F_{4} (x) F_{81} (x) \\ F_{81} (x) & = F_{82} (x) + F_{85} (x) \\ F_{82} (x) & = F_{76} (x) + F_{83} (x) \\ F_{83} (x) & = F_{84} (x) \\ F_{84} (x) & = F_{4} (x) F_{43} (x) \\ F_{85} (x) & = F_{79} (x) + F_{86} (x) \\ F_{86} (x) & = 2 F_{15} (x) + F_{60} (x) + F_{61} (x) + F_{87} (x) + F_{91} (x) \\ F_{87} (x) & = F_{4} (x) F_{88} (x) \\ F_{88} (x) & = F_{89} (x) + F_{90} (x) \\ F_{89} (x) & = F_{83} (x) \\ F_{90} (x) & = F_{86} (x) \\ F_{91} (x) & = F_{4} (x) F_{64} (x) \\ F_{92} (x) & = 0 \\ F_{93} (x) & = F_{4} (x) F_{94} (x) \\ F_{94} (x) & = F_{95} (x) \\ F_{95} (x) & = F_{113} (x) + F_{142} (x) + F_{15} (x) + F_{96} (x) \\ F_{96} (x) & = F_{4} (x) F_{97} (x) \\ F_{97} (x) & = F_{100} (x) + F_{98} (x) \\ F_{98} (x) & = F_{14} (x) + F_{99} (x) \\ F_{99} (x) & = F_{15} (x) + F_{33} (x) + F_{40} (x) + F_{77} (x) \\ F_{100} (x) & = F_{101} (x) + F_{95} (x) \\ F_{101} (x) & = F_{102} (x) + F_{15} (x) + F_{71} (x) + F_{92} (x) + F_{93} (x) \\ F_{102} (x) & = F_{103} (x) F_{4} (x) \\ F_{103} (x) & = F_{104} (x) + F_{106} (x) \\ F_{104} (x) & = F_{105} (x) + F_{99} (x) \\ F_{105} (x) & = F_{84} (x) \\ F_{106} (x) & = F_{101} (x) + F_{107} (x) \\ F_{107} (x) & = F_{108} (x) + F_{112} (x) + F_{15} (x) + F_{60} (x) + F_{61} (x) + F_{91} (x) \\ F_{108} (x) & = F_{109} (x) F_{4} (x) \\ F_{109} (x) & = F_{110} (x) + F_{111} (x) \\ F_{110} (x) & = F_{105} (x) \\ F_{111} (x) & = F_{107} (x) \\ F_{112} (x) & = 0 \\ F_{113} (x) & = F_{114} (x) F_{4} (x) \\ F_{114} (x) & = F_{115} (x) \\ F_{115} (x) & = F_{116} (x) + F_{15} (x) + F_{175} (x) \\ F_{116} (x) & = F_{117} (x) F_{4} (x) \\ F_{117} (x) & = F_{118} (x) + F_{120} (x) \\ F_{118} (x) & = F_{119} (x) + F_{4} (x) \\ F_{119} (x) & = F_{15} (x) + F_{18} (x) + F_{25} (x) \\ F_{120} (x) & = F_{115} (x) + F_{121} (x) \\ F_{121} (x) & = F_{122} (x) + F_{140} (x) + F_{142} (x) + F_{15} (x) \\ F_{122} (x) & = F_{123} (x) F_{4} (x) \\ F_{123} (x) & = F_{124} (x) + F_{126} (x) \\ F_{124} (x) & = F_{119} (x) + F_{125} (x) \\ F_{125} (x) & = F_{15} (x) + F_{33} (x) + F_{40} (x) + F_{41} (x) \\ F_{126} (x) & = F_{121} (x) + F_{127} (x) \\ F_{127} (x) & = F_{128} (x) + F_{138} (x) + F_{15} (x) + F_{71} (x) + F_{92} (x) \\ F_{128} (x) & = F_{129} (x) F_{4} (x) \\ F_{129} (x) & = F_{130} (x) + F_{132} (x) \\ F_{130} (x) & = F_{125} (x) + F_{131} (x) \\ F_{131} (x) & = F_{52} (x) \\ F_{132} (x) & = F_{127} (x) + F_{133} (x) \\ F_{133} (x) & = F_{112} (x) + F_{134} (x) + F_{15} (x) + F_{60} (x) + F_{61} (x) + F_{62} (x) \\ F_{134} (x) & = F_{135} (x) F_{4} (x) \\ F_{135} (x) & = F_{136} (x) + F_{137} (x) \\ F_{136} (x) & = F_{131} (x) \\ F_{137} (x) & = F_{133} (x) \\ F_{138} (x) & = F_{139} (x) F_{4} (x) \\ F_{139} (x) & = F_{64} (x) + F_{94} (x) \\ F_{140} (x) & = F_{141} (x) F_{4} (x) \\ F_{141} (x) & = F_{114} (x) + F_{94} (x) \\ F_{142} (x) & = F_{143} (x) F_{4} (x) \\ F_{143} (x) & = F_{144} (x) + F_{15} (x) + F_{173} (x) \\ F_{144} (x) & = F_{145} (x) F_{4} (x) \\ F_{145} (x) & = F_{146} (x) + F_{149} (x) \\ F_{146} (x) & = F_{12} (x) + F_{147} (x) \\ F_{147} (x) & = F_{148} (x) + F_{15} (x) + F_{25} (x) \\ F_{148} (x) & = F_{11} (x) F_{4} (x) \\ F_{149} (x) & = F_{143} (x) + F_{150} (x) \\ F_{150} (x) & = F_{140} (x) + F_{15} (x) + F_{151} (x) + F_{171} (x) \\ F_{151} (x) & = F_{152} (x) F_{4} (x) \\ F_{152} (x) & = F_{153} (x) + F_{156} (x) \\ F_{153} (x) & = F_{147} (x) + F_{154} (x) \\ F_{154} (x) & = F_{15} (x) + F_{155} (x) + F_{40} (x) + F_{41} (x) \\ F_{155} (x) & = F_{30} (x) F_{4} (x) \\ F_{156} (x) & = F_{150} (x) + F_{157} (x) \\ F_{157} (x) & = F_{138} (x) + F_{15} (x) + F_{158} (x) + F_{169} (x) + F_{92} (x) \\ F_{158} (x) & = F_{159} (x) F_{4} (x) \\ F_{159} (x) & = F_{160} (x) + F_{162} (x) \\ F_{160} (x) & = F_{154} (x) + F_{161} (x) \\ F_{161} (x) & = F_{52} (x) \\ F_{162} (x) & = F_{157} (x) + F_{163} (x) \\ F_{163} (x) & = F_{15} (x) + F_{164} (x) + F_{168} (x) + F_{60} (x) + F_{61} (x) + F_{62} (x) \\ F_{164} (x) & = F_{165} (x) F_{4} (x) \\ F_{165} (x) & = F_{166} (x) + F_{167} (x) \\ F_{166} (x) & = F_{161} (x) \\ F_{167} (x) & = F_{163} (x) \\ F_{168} (x) & = 0 \\ F_{169} (x) & = F_{170} (x) F_{4} (x) \\ F_{170} (x) & = F_{65} (x) + F_{72} (x) \\ F_{171} (x) & = F_{172} (x) F_{4} (x) \\ F_{172} (x) & = F_{143} (x) + F_{95} (x) \\ F_{173} (x) & = F_{174} (x) F_{4} (x) \\ F_{174} (x) & = F_{115} (x) + F_{2} (x) \\ F_{175} (x) & = F_{2} (x) F_{4} (x) \\ F_{176} (x) & = 0 \\ F_{177} (x) & = F_{178} (x) F_{4} (x) \\ F_{178} (x) & = F_{170} (x) \\ F_{179} (x) & = F_{180} (x) F_{4} (x) \\ F_{180} (x) & = F_{170} (x) + F_{172} (x) \\ F_{181} (x) & = F_{182} (x) F_{4} (x) \\ F_{182} (x) & = F_{172} (x) + F_{174} (x) \end{aligned}

Av(1234, 1243, 1432)

This specification was found using the strategy pack "Point Placements" and has 183 rules.

Proof Tree

System of Equations