length 2: 12, 21
length 3: 123, 132, 213, 231, 312, 321
length 4: 1234, 2134, 2314, 2341, 3124, 3214, 3241, 3421, 4123, 4213, 4231, 4312, 4321
length 5: 12345, 21345, 23145, 23415, 23451, 31245, 32145, 32415, 32451, 34215, 34251, 34521, 41235, 42135, 42315, 42351, 43125, 43215, 43251, 43521, 45321, 51234, 52134, 52314, 52341, 53124, 53214, 53241, 53421, 54123, 54213, 54231, 54312, 54321
$\mathcal{A}$ = Av(1243, 1324, 1342, 1423, 1432, 2143, 2413, 2431, 3142, 3412, 4132)
$\mathcal{B}$ = Av(132, 213, 231, 312, 321)
$\operatorname{F_{6}}{\left (x \right )} = \operatorname{F_{0}}{\left (x \right )} + \operatorname{F_{5}}{\left (x \right )}$
$\operatorname{F_{0}}{\left (x \right )} = 1$
$\operatorname{F_{5}}{\left (x \right )} = \operatorname{F_{1}}{\left (x \right )} + \operatorname{F_{4}}{\left (x \right )}$
$\operatorname{F_{1}}{\left (x \right )} = \frac{x \left(- 2 x + 1\right)}{x^{2} - 3 x + 1}$
$\operatorname{F_{4}}{\left (x \right )} = \operatorname{F_{2}}{\left (x \right )} + \operatorname{F_{3}}{\left (x \right )}$
$\operatorname{F_{2}}{\left (x \right )} = \frac{x^{2}}{x^{2} - 3 x + 1}$
$\operatorname{F_{3}}{\left (x \right )} = x^{3}$
$\mathcal{A}$ |
$\bullet$ | ||
$\bullet$ | ||
$\bullet$ |
$\mathcal{B}$ |
$\mathcal{A}$ = Av(1243, 1324, 1342, 1423, 1432, 2143, 2413, 2431, 3142, 3412, 4132)
$a_{0} = 1,~a_{1} = 1,~a_{2} = 2,~a_{3} = 6$
$a_{n} = b_{n}$