Mathematics – Combinatorics
Scientific paper
2008-05-12
Mathematics
Combinatorics
7 pages
Scientific paper
The main result of the note is a combinatorial identity that expresses the partition's quantity of natural $n$ with $q$ distinct parts by means of the partitions of $n$, for which the differences between parts are not less than either $\lb=2$, or $\lb=3$. Such partitions are called $\lb$-partitions. For them is introduced a notion of index - a non negative integer that depends on $\lb$. One corollary of the identity is the formula $d(n)=\sum_{\alpha=0}^\infty p_\lb(n,\alpha)2^\alpha$, where $d(n)$ is the partition's quantity of $n$ with distinct parts and $p_\lb(n,\alpha)$ is the $\lb$-partition's quantity of $n$, index of which equals to $\alpha$. For $\lb=3$ the identity turns to be equivalent to the famous Sylvester formula and gives a new combinatorial interpretation for it. Two bijective proofs of the main result are provided: one for $\lb=3$ and another one for $\lb=2$ and $\lb=3$ simultaneously.
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