3 problems

Schur Q-positivity conjecture. For any increasing permutation $$ , the pattern-avoiding peak function $R_n$ is symmetric and Schur $Q$-positive for all $n\geq 0$.

Peak-idempotent decomposition conjecture. For any odd partition $mu$ with $|mu|\leq n$ one has

Let $\PCp$ denote the set of peak compositions, and for each peak composition $\alpha$ let $K_\alpha$ be the corresponding peak quasisymmetric function. The quasisymmetric Schur…