Inversions in a permutation $Y$ are defined as pairs where $Y_a < Y_b$ but $a > b$, while fixed points in $Y$ are defined as elements where $Y_a = a$ (i.e., 1-cycles). Let $S_\alpha$ be the set of all permutations of length $n$ with exactly $j$ inversions and $k$ fixed points, for some $j \leq n$ and some possible $k \leq n$ (i.e., not all $k$ are possible for all $j$). Let $S_\beta$ be the set of all permutations of length $n$ with exactly $j-1$ inversions. Let $S_\delta$ be the subset of $S_\beta$ with fewer than $k$ fixed points. I conjecture the cardinality of $S_\delta$ is no greater than half the cardinality of $S_\beta$, for all $n > 1$, all $j$, and all possible $k$.

The significance of the conjecture is that, if true, it implies the expected number of fixed points strictly increases as the expected number of inversions decreases. That, in turn, implies the expected number of fixed points in a bivariate random sample $(X, Y)$, from a population distribution with correlation parameter $\rho$, is monotone increasing in $\rho$, for any distribution where expected Kendall's $\tau$ is increasing in $\rho$ (e.g., the bivariate normal).

Edit: The "implication" stated above is false. The conjecture is strictly combinatorial--it assumes all permutations occur with equal frequency--so it can have no direct implication under nonzero tau--where permutations nearer the uniform permutation (1, 2, ... , n) occur with greater frequency as tau becomes more positive.

I'm looking for a proof if one exists (a direct proof of either implication would do as well). I've searched the mathematical statistics literature with little luck--although Mallows models with the Kendall tau distance seem promising--but I'm not a combinatorialist. Otherwise, thoughts on how to proceed are appreciated.