  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 $k\geqslant 1$.
