Two combinatorial formulas concerning marked partitions

A partition of degree $n$ is a decomposition $n=i_1+i_2+\dots+i_q$, where ${i_1,i_2,\dots,i_q}$ are positive integers called the parts of the partition. Let $\lambda>0$ be an integer. The partition is said to be a $\lambda$--partition if $i_{a+1}-i_a\geqslant \lambda$ for all $a$ such that $1\leqslant a<q$. The main result of this note are combinatorial formulas, which express the quantity of $1$-partitions of a given degree in terms of the $\lambda$--partitions of the same degree, where $\lambda=2$ or $\lambda=3$, some special parts of which are marked depending on $\lambda$. The presented proofs of both formulas are bijective. It is shown that for $\lambda=3$ the corresponding formula is equivalent to the classical Sylvester identity. The obtained combinatorial formulas as well as their bijective proofs are generalized to the quantities of $1$--partitions, all parts of which are $\geqslant k$ for any fixed integer $k\geqslant 1$.