Av(12345, 12354, 12435, 13245, 13254, 13425, 14235, 14325, 23145, 23154, 23415, 24135, 24315, 34125, 34215)
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Counting Sequence
1, 1, 2, 6, 24, 105, 480, 2264, 10940, 53891, 269650, 1366754, 7002992, 36214028, 188760920, ...
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Implicit Equation for the Generating Function
\(\displaystyle x^{2} \left(x -2\right) F \left(x \right)^{3}+x \left(x^{2}-2 x +8\right) F \left(x \right)^{2}+\left(-3 x^{3}-9 x -2\right) F \! \left(x \right)+x^{3}+x^{2}+3 x +2 = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 24\)
\(\displaystyle a \! \left(5\right) = 105\)
\(\displaystyle a \! \left(6\right) = 480\)
\(\displaystyle a \! \left(7\right) = 2264\)
\(\displaystyle a \! \left(8\right) = 10940\)
\(\displaystyle a \! \left(n +9\right) = \frac{n \left(n +1\right) a \! \left(n \right)}{2 \left(n +10\right) \left(n +9\right)}-\frac{\left(28 n +57\right) \left(n +1\right) a \! \left(n +1\right)}{8 \left(n +10\right) \left(n +9\right)}+\frac{\left(213 n^{2}+48 n -989\right) a \! \left(n +2\right)}{64 \left(n +10\right) \left(n +9\right)}-\frac{\left(1451 n^{2}+3233 n -2478\right) a \! \left(n +3\right)}{384 \left(n +10\right) \left(n +9\right)}+\frac{\left(1304 n^{2}+2057 n -23145\right) a \! \left(n +4\right)}{192 \left(n +10\right) \left(n +9\right)}+\frac{\left(2365 n^{2}+25891 n +71898\right) a \! \left(n +5\right)}{96 \left(n +10\right) \left(n +9\right)}-\frac{\left(251 n^{2}+2478 n +5163\right) a \! \left(n +6\right)}{16 \left(n +10\right) \left(n +9\right)}-\frac{\left(272 n^{2}+4661 n +19851\right) a \! \left(n +7\right)}{24 \left(n +10\right) \left(n +9\right)}+\frac{\left(97 n +829\right) a \! \left(n +8\right)}{12 n +120}, \quad n \geq 9\)
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This specification was found using the strategy pack "Col Placements Tracked Fusion" and has 18 rules.

Found on January 23, 2022.

Finding the specification took 13 seconds.

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\(\begin{align*} F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{1}\! \left(x \right) &= 1\\ F_{2}\! \left(x \right) &= F_{3}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{3}\! \left(x \right) &= x\\ F_{4}\! \left(x \right) &= F_{5}\! \left(x , 1\right)\\ F_{5}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{6}\! \left(x , y\right)+F_{8}\! \left(x , y\right)\\ F_{6}\! \left(x , y\right) &= F_{3}\! \left(x \right) F_{7}\! \left(x , y\right)\\ F_{7}\! \left(x , y\right) &= \frac{F_{5}\! \left(x , y\right) y -F_{5}\! \left(x , 1\right)}{-1+y}\\ F_{8}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{9}\! \left(x , y\right)\\ F_{9}\! \left(x , y\right) &= y x\\ F_{10}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{11}\! \left(x , y\right)+F_{14}\! \left(x , y\right)+F_{16}\! \left(x , y\right)\\ F_{11}\! \left(x , y\right) &= F_{12}\! \left(x , y\right)\\ F_{12}\! \left(x , y\right) &= F_{13}\! \left(x , y\right) F_{3}\! \left(x \right) F_{7}\! \left(x , y\right)\\ F_{13}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{9}\! \left(x , y\right)\\ F_{14}\! \left(x , y\right) &= F_{15}\! \left(x , y\right)\\ F_{15}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{13}\! \left(x , y\right) F_{9}\! \left(x , y\right)\\ F_{16}\! \left(x , y\right) &= F_{17}\! \left(x , y\right)\\ F_{17}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{13}\! \left(x , y\right) F_{9}\! \left(x , y\right)\\ \end{align*}\)

This specification was found using the strategy pack "Col Placements Tracked Fusion" and has 19 rules.

Found on January 22, 2022.

Finding the specification took 21 seconds.

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\(\begin{align*} F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{1}\! \left(x \right) &= 1\\ F_{2}\! \left(x \right) &= F_{3}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{3}\! \left(x \right) &= x\\ F_{4}\! \left(x \right) &= F_{5}\! \left(x , 1\right)\\ F_{5}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{6}\! \left(x , y\right)+F_{8}\! \left(x , y\right)\\ F_{6}\! \left(x , y\right) &= F_{3}\! \left(x \right) F_{7}\! \left(x , y\right)\\ F_{7}\! \left(x , y\right) &= -\frac{-y F_{5}\! \left(x , y\right)+F_{5}\! \left(x , 1\right)}{-1+y}\\ F_{8}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{9}\! \left(x , y\right)\\ F_{9}\! \left(x , y\right) &= y x\\ F_{10}\! \left(x , y\right) &= F_{11}\! \left(x , 1, y\right)\\ F_{11}\! \left(x , y , z\right) &= F_{1}\! \left(x \right)+F_{12}\! \left(x , y , z\right)+F_{15}\! \left(x , y , z\right)+F_{17}\! \left(x , z , y\right)\\ F_{12}\! \left(x , y , z\right) &= F_{13}\! \left(x , z , y\right)\\ F_{13}\! \left(x , y , z\right) &= F_{14}\! \left(x , y\right) F_{3}\! \left(x \right) F_{7}\! \left(x , z\right)\\ F_{14}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{9}\! \left(x , y\right)\\ F_{15}\! \left(x , y , z\right) &= F_{16}\! \left(x , z , y\right)\\ F_{16}\! \left(x , y , z\right) &= F_{10}\! \left(x , z\right) F_{14}\! \left(x , y\right) F_{9}\! \left(x , z\right)\\ F_{17}\! \left(x , y , z\right) &= F_{18}\! \left(x , y , z\right)\\ F_{18}\! \left(x , y , z\right) &= F_{11}\! \left(x , z , y\right) F_{14}\! \left(x , y\right) F_{9}\! \left(x , y\right)\\ \end{align*}\)