cycle as the starting point and then arranging the cycles with increasing maximum elements.

We are primarily interested in $$W(N,k) = \sum_{k\text{-winnable } \pi \in {\mathfrak{S}}_N} {\theta}^{\#\text{left-to-right maxima in } \pi}.$$ Here, we say that $\pi$ is [**$k$-winnable**]{} if it would be won by the positional strategy that rejects the first $k$ candidates and accepts the next left-to-right maximum thereafter. Some examples of these polynomials are given in Figure \[f:W\]. Our next result provides a recursive description for them.

\[t:W\] We have $$W(N,k) = (N-1) W(N-1,k) + \frac{(N-2)!}{(k-1)!} {\theta}{\{k\}!}$$ with initial conditions $W(1, 0) = {\theta}$ and $W(N,N) = 0$.

We have two cases for the $k$-winnable permutations $\pi \in {\mathfrak{S}}_{N}$.

-   If the last position contains one of the values $1, 2, \ldots, N-1$, then it is not a left-to-right maximum and we may view the complementary values as some $k$-winnable $\widetilde{\pi} \in {\mathfrak{S}}_{N-1}$. Hence, these contribute $(N-1) W(N-1,k)$ to $W(N,k)$.

-   If the last position contains $N$ then it is a left-to-right maximum and the value $N-1$ must lie in one of the first $k$ positions in order for $\pi$ to be $k$-winnable. We can choose the rest of the values to place among the first $k$ positions 