Let $P(n,m)$ denote the set of all positive integer partitions of $n$ into parts that are pairwise distinct and bounded by $m$. Let $p(n,m) = |P(n,m)|$.

After some numerical experiments it appears

$$ p(n,m) + p(n+m,m) \geq p(n+k,m) $$

for all $1 \leq k \leq m$.

(An even sharper inequality may hold, but the above would be sufficient for the purposes of my investigation).

Would it be possible to explain this via an injection

$\sigma: P(n+k,m) \to P(n,m)\cup P(n+m,m)$?

Although less preferable, a proof (or reference to a proof) by other techniques would also be accepted.