PermPal Database

Av(1234, 1243, 3412)

Generating Function

- \frac{12 x^{7} - 36 x^{6} + 78 x^{5} - 98 x^{4} + 75 x^{3} - 35 x^{2} + 9 x - 1}{{(2 x - 1)}^{2} {(x - 1)}^{6}}

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Counting Sequence

1, 1, 2, 6, 21, 73, 238, 718, 2013, 5301, 13266, 31886, 74269, 168841, 376750, ...

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Implicit Equation for the Generating Function

{(2 x - 1)}^{2} {(x - 1)}^{6} F (x) + 12 x^{7} - 36 x^{6} + 78 x^{5} - 98 x^{4} + 75 x^{3} - 35 x^{2} + 9 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) = 73

a (6) = 238

a (7) = 718

a (n + 2) = - \frac{n^{5}}{30} + \frac{5 n^{4}}{12} - \frac{4 n^{3}}{3} + \frac{37 n^{2}}{12} - 4 a (n) + 4 a (n + 1) - \frac{32 n}{15} + 2, n \geq 8

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Explicit Closed Form

5 + 2^{n + 1} n - 4 2^{n} - \frac{2 n^{3}}{3} + \frac{5 n^{2}}{12} + \frac{n}{5} + \frac{n^{4}}{12} - \frac{n^{5}}{30}

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This specification was found using the strategy pack "Insertion Point Placements" and has 142 rules.

Found on January 17, 2022.

Finding the specification took 33 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_{22} (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_{7} (x) \\ F_{12} (x) & = F_{13} (x) + F_{14} (x) + F_{18} (x) \\ F_{13} (x) & = 0 \\ F_{14} (x) & = F_{15} (x) F_{4} (x) \\ F_{15} (x) & = F_{16} (x) + F_{17} (x) \\ F_{16} (x) & = F_{4} (x) \\ F_{17} (x) & = F_{12} (x) \\ F_{18} (x) & = F_{19} (x) F_{4} (x) \\ F_{19} (x) & = F_{20} (x) \\ F_{20} (x) & = F_{21} (x) F_{4} (x) \\ F_{21} (x) & = F_{1} (x) + F_{20} (x) \\ F_{22} (x) & = F_{2} (x) + F_{23} (x) \\ F_{23} (x) & = F_{24} (x) \\ F_{24} (x) & = F_{25} (x) F_{4} (x) \\ F_{25} (x) & = F_{103} (x) + F_{26} (x) \\ F_{26} (x) & = F_{27} (x) + F_{28} (x) \\ F_{27} (x) & = F_{23} (x) + F_{7} (x) \\ F_{28} (x) & = F_{29} (x) + F_{55} (x) \\ F_{29} (x) & = F_{30} (x) + F_{7} (x) \\ F_{30} (x) & = F_{13} (x) + F_{31} (x) + F_{37} (x) \\ F_{31} (x) & = F_{32} (x) F_{4} (x) \\ F_{32} (x) & = F_{33} (x) + F_{34} (x) \\ F_{33} (x) & = F_{4} (x) \\ F_{34} (x) & = F_{35} (x) \\ F_{35} (x) & = F_{13} (x) + F_{31} (x) + F_{36} (x) \\ F_{36} (x) & = F_{19} (x) F_{4} (x) \\ F_{37} (x) & = F_{38} (x) F_{4} (x) \\ F_{38} (x) & = F_{39} (x) + F_{43} (x) \\ F_{39} (x) & = F_{19} (x) + F_{40} (x) \\ F_{40} (x) & = F_{13} (x) + F_{36} (x) + F_{41} (x) \\ F_{41} (x) & = F_{4} (x) F_{42} (x) \\ F_{42} (x) & = F_{4} (x) + F_{40} (x) \\ F_{43} (x) & = F_{44} (x) + F_{45} (x) \\ F_{44} (x) & = F_{13} (x) + F_{37} (x) + F_{41} (x) \\ F_{45} (x) & = F_{13} (x) + F_{46} (x) + F_{47} (x) + F_{51} (x) \\ F_{46} (x) & = 0 \\ F_{47} (x) & = F_{4} (x) F_{48} (x) \\ F_{48} (x) & = F_{49} (x) + F_{50} (x) \\ F_{49} (x) & = F_{40} (x) \\ F_{50} (x) & = F_{45} (x) \\ F_{51} (x) & = F_{4} (x) F_{52} (x) \\ F_{52} (x) & = F_{53} (x) \\ F_{53} (x) & = F_{4} (x) F_{54} (x) \\ F_{54} (x) & = F_{19} (x) + F_{52} (x) \\ F_{55} (x) & = F_{56} (x) + F_{57} (x) \\ F_{56} (x) & = F_{19} (x) F_{7} (x) \\ F_{57} (x) & = F_{58} (x) \\ F_{58} (x) & = F_{4} (x) F_{59} (x) \\ F_{59} (x) & = F_{60} (x) + F_{70} (x) \\ F_{60} (x) & = F_{61} (x) \\ F_{61} (x) & = F_{19} (x) F_{21} (x) F_{62} (x) \\ F_{62} (x) & = F_{63} (x) + F_{68} (x) \\ F_{63} (x) & = F_{19} (x) + F_{64} (x) \\ F_{64} (x) & = F_{13} (x) + F_{65} (x) + F_{67} (x) \\ F_{65} (x) & = F_{4} (x) F_{66} (x) \\ F_{66} (x) & = F_{19} (x) + F_{64} (x) \\ F_{67} (x) & = F_{4} (x) F_{63} (x) \\ F_{68} (x) & = F_{19} (x) F_{69} (x) \\ F_{69} (x) & = F_{21} (x) + F_{63} (x) \\ F_{70} (x) & = F_{19} (x) F_{71} (x) \\ F_{71} (x) & = F_{60} (x) + F_{72} (x) \\ F_{72} (x) & = F_{73} (x) + F_{98} (x) \\ F_{73} (x) & = F_{74} (x) + F_{96} (x) \\ F_{74} (x) & = F_{19} (x) + F_{75} (x) \\ F_{75} (x) & = F_{13} (x) + F_{76} (x) + F_{78} (x) \\ F_{76} (x) & = F_{4} (x) F_{77} (x) \\ F_{77} (x) & = F_{7} (x) + F_{75} (x) \\ F_{78} (x) & = F_{4} (x) F_{79} (x) \\ F_{79} (x) & = F_{80} (x) + F_{84} (x) \\ F_{80} (x) & = F_{19} (x) + F_{81} (x) \\ F_{81} (x) & = F_{13} (x) + F_{18} (x) + F_{82} (x) \\ F_{82} (x) & = F_{4} (x) F_{83} (x) \\ F_{83} (x) & = F_{4} (x) + F_{81} (x) \\ F_{84} (x) & = F_{75} (x) + F_{85} (x) \\ F_{85} (x) & = F_{13} (x) + F_{86} (x) + F_{88} (x) + F_{92} (x) \\ F_{86} (x) & = F_{4} (x) F_{87} (x) \\ F_{87} (x) & = F_{12} (x) + F_{85} (x) \\ F_{88} (x) & = F_{4} (x) F_{89} (x) \\ F_{89} (x) & = F_{90} (x) + F_{91} (x) \\ F_{90} (x) & = F_{81} (x) \\ F_{91} (x) & = F_{85} (x) \\ F_{92} (x) & = F_{4} (x) F_{93} (x) \\ F_{93} (x) & = F_{94} (x) \\ F_{94} (x) & = F_{4} (x) F_{95} (x) \\ F_{95} (x) & = F_{19} (x) + F_{93} (x) \\ F_{96} (x) & = F_{97} (x) \\ F_{97} (x) & = F_{19} {(x)}^{2} F_{21} (x) \\ F_{98} (x) & = F_{101} (x) + F_{99} (x) \\ F_{99} (x) & = F_{100} (x) F_{19} (x) \\ F_{100} (x) & = F_{6} (x) + F_{74} (x) \\ F_{101} (x) & = F_{102} (x) \\ F_{102} (x) & = F_{21} {(x)}^{2} F_{64} (x) \\ F_{103} (x) & = F_{104} (x) + F_{132} (x) \\ F_{104} (x) & = F_{105} (x) + F_{106} (x) \\ F_{105} (x) & = F_{19} (x) F_{4} (x) \\ F_{106} (x) & = F_{107} (x) \\ F_{107} (x) & = F_{108} (x) F_{4} (x) \\ F_{108} (x) & = F_{109} (x) + F_{117} (x) \\ F_{109} (x) & = F_{110} (x) + F_{111} (x) \\ F_{110} (x) & = F_{104} (x) \\ F_{111} (x) & = F_{112} (x) F_{4} (x) \\ F_{112} (x) & = F_{113} (x) + F_{19} (x) \\ F_{113} (x) & = F_{114} (x) + F_{116} (x) + F_{13} (x) \\ F_{114} (x) & = F_{115} (x) F_{4} (x) \\ F_{115} (x) & = F_{113} (x) + F_{19} (x) \\ F_{116} (x) & = F_{112} (x) F_{4} (x) \\ F_{117} (x) & = F_{118} (x) F_{19} (x) \\ F_{118} (x) & = F_{119} (x) + F_{123} (x) \\ F_{119} (x) & = F_{120} (x) + F_{20} (x) \\ F_{120} (x) & = F_{121} (x) + F_{122} (x) \\ F_{121} (x) & = F_{12} (x) + F_{4} (x) \\ F_{122} (x) & = F_{81} (x) + F_{85} (x) \\ F_{123} (x) & = F_{111} (x) + F_{124} (x) \\ F_{124} (x) & = F_{125} (x) \\ F_{125} (x) & = F_{126} (x) F_{4} (x) \\ F_{126} (x) & = F_{112} (x) + F_{127} (x) \\ F_{127} (x) & = F_{113} (x) + F_{128} (x) \\ F_{128} (x) & = 2 F_{13} (x) + F_{129} (x) + F_{131} (x) \\ F_{129} (x) & = F_{130} (x) F_{4} (x) \\ F_{130} (x) & = F_{128} (x) + F_{52} (x) \\ F_{131} (x) & = F_{127} (x) F_{4} (x) \\ F_{132} (x) & = F_{133} (x) + F_{135} (x) \\ F_{133} (x) & = F_{134} (x) F_{19} (x) \\ F_{134} (x) & = F_{35} (x) + F_{7} (x) \\ F_{135} (x) & = F_{136} (x) + F_{137} (x) \\ F_{136} (x) & = F_{113} (x) F_{7} (x) \\ F_{137} (x) & = F_{138} (x) \\ F_{138} (x) & = F_{139} (x) F_{140} (x) F_{4} (x) \\ F_{139} (x) & = F_{113} (x) + F_{127} (x) \\ F_{140} (x) & = F_{141} (x) + F_{19} (x) \\ F_{141} (x) & = F_{19} (x) F_{21} (x) \end{aligned}

Av(1234, 1243, 3412)

This specification was found using the strategy pack "Insertion Point Placements" and has 142 rules.

Proof Tree

System of Equations