This question is about symmetric functions. As usual, let $s_\lambda$ denote the Schur function corresponding to a partition $\lambda$. For $r\geqslant1$, let $d_r$ denote the map defined on symmetric functions by mapping $s_\lambda\mapsto s_{\lambda/(1^r)}$ and extending linearly. In other words, $s_\lambda\mapsto\sum_\mu s_\mu$, summed over all partitions $\mu$ obtained from $\lambda$ by removing $r$ nodes in distinct rows.

Let $P_\lambda$ denote the Schur P-function corresponding to the strict partition $\lambda$. I would like to write $d_rP_\lambda$ in terms of Schur P-functions, and it appears that

$ d_rP_\lambda = \sum_\mu2^{j(\lambda,\mu)}P_\mu, $

summed over all strict partitions $\mu$ obtained from $\lambda$ by removing $r$ nodes in distinct columns; $j(\lambda,\mu)$ denotes the number of $i\geqslant2$ such that there is a node of $\lambda/\mu$ in column $i$ but not in column $i-1$.

I expect there is a generalisation to Hall-Littlewood functions, though for now I only need this special case. This generalisation appears to be in some sense dual to the Pieri-type rule for Hall-Littlewood functions (III.5.7 in Macdonald), but I don't know enough about duality to do this kind of thing.

Is my conjectured formula in the literature, or can it be easily deduced from known results?

**Update:** I can now prove my formula, using marked tableaux and the fact that the Schur P-functions are invariant under the involution $\omega$. But it would be good to know if this is already known or easy to prove.

**Another update (27th February 2021)** I can now prove a version for Hall-Littlewood functions using duality. Please e-mail me if you would like to see the details.