Reselection in the game of best choice

We investigate a remarkable probability distribution on the symmetric group, due to Steck from the early 1970's, arising from a natural process that intertwines continuous and discrete selections for the values and positions, respectively, of a permutation. Steck used a matrix determinant to express his distribution, whereas we contribute new combinatorial formulas for it in terms of "bottom-to-top maxima" (that are simply the left-to-right maxima of the inverse) permutation statistics. These formulas specialize, in the case of the identity permutation, to a result of Pitman--Stanley from the late 1990's. We then use the Steck distribution to define a game of best choice (secretary problem variation) that incorporates a filtering process for the pool of candidates over time. We solve the model for the case where there is a single filtering step. It turns out that the probability of winning the game under optimal play is similar to the classical model ($1/e$, asymptotically), but that the interviewer must employ a different (non-positional) strategy in order to attain it. The optimal strategy depends on the relationship between interview position and the next bottom-to-top maximum value after the filtering step. Among other results, we prove that the optimal strategy always transitions from rejection to acceptance between positions $(1/e)$ and $(1/e) + (1 - 1/e)y$ as a proportion of the total candidates considered, where $y$ is the proportion of unfiltered candidates.