We denote by $S_{n}$ the set of permutations of $\llbracket 1, n \rrbracket$. We represent an element $\sigma$ of $S_{n}$ by the finite sequence $(\sigma(1), \sigma(2), \ldots, \sigma(n))$ and we call an ascent (respectively descent) of $\sigma$ an ascent of this sequence. For any integer $m$, we denote by $A_{n,m}$ the number of elements of $S_{n}$ with $m$ ascents. In Friedman's urn model ($a_{0} = 1, b_{0} = 0, a = d = 0, b = c = 1$), $P_{3}(u,v) = uv^{3} + 4u^{2}v^{2} + u^{3}v$. Determine the elements of $S_{3}$ and calculate among them the number of permutations with $m$ ascents for any integer $m$. Compare the values obtained with the coefficients of $P_{3}(X, 1)$ where $P_{3}$ was expressed in question 13.
We denote by $S_{n}$ the set of permutations of $\llbracket 1, n \rrbracket$. We represent an element $\sigma$ of $S_{n}$ by the finite sequence $(\sigma(1), \sigma(2), \ldots, \sigma(n))$ and we call an ascent (respectively descent) of $\sigma$ an ascent of this sequence. For any integer $m$, we denote by $A_{n,m}$ the number of elements of $S_{n}$ with $m$ ascents.
In Friedman's urn model ($a_{0} = 1, b_{0} = 0, a = d = 0, b = c = 1$), $P_{3}(u,v) = uv^{3} + 4u^{2}v^{2} + u^{3}v$.
Determine the elements of $S_{3}$ and calculate among them the number of permutations with $m$ ascents for any integer $m$. Compare the values obtained with the coefficients of $P_{3}(X, 1)$ where $P_{3}$ was expressed in question 13.