q-analogue for Richardson tableaux with a prescribed number of odd columns

Prove the conjectured q-enumeration of Richardson tableaux of size n with k odd columns: for k congruent to n modulo 2, establish that the sum of q raised to the comajor index over these tableaux equals q raised to the binomial coefficient of k choose 2 times the q-binomial coefficient of n choose k times the q-Catalan number of (n-k)/2.

Background

The paper establishes an ordinary enumeration of Richardson tableaux with a prescribed number k of odd columns, identifying the count with a binomial coefficient multiplied by a Catalan number. It also proves q-counting formulas for even Richardson tableaux, corresponding to noncrossing perfect matchings and q-Catalan numbers.

The authors then propose a broader q-analogue for all Richardson tableaux with a fixed number of odd columns. The conjecture has been checked computationally for n up to 14 and specializes to the proved even-tableau formula when k=0 and n is replaced by 2n.