This site is supported by donations to The OEIS Foundation.

I'm teacher of math in secondary school and my primary interest is combinatorics especially theory of compositions and partitions of natural numbers and sets.

## Note on notations

(1)......${\displaystyle N=\{0,1,2,...\}\,}$ the set of natural numbers
(2)......${\displaystyle I_{a}^{b}=\{x:b\leq x
(3)......${\displaystyle I_{a}^{0}=I_{a}\,}$
(4)......${\displaystyle I_{a+1}^{1}=N_{a}\,}$
(5)......${\displaystyle I_{\infty }^{b}=I^{b}\,}$
(6)......${\displaystyle O=\{x:x=2n+1,n\in N\}\,}$
(7)......${\displaystyle E=\{x:x=2n,n\in N\}\,}$

## Composition of natural number k over set S

${\displaystyle c_{m}(k,S)=\sum _{\stackrel {c_{0}+c_{1}+...+c_{m-1}=k}{c_{i}\in S\cap I_{k+1},i\in I_{m}}}1}$

## Partitions of natural number k over set S

${\displaystyle p_{m}(k,S)=\sum _{\stackrel {t_{0}+t_{1}+...+t_{k}=m}{\begin{matrix}\scriptstyle t_{1}+2t_{2}+...+kt_{k}=k\\\scriptstyle t_{i}=0,i\notin S\cap I_{k+1}\end{matrix}}}1\,}$

## Composition of a k-set over set S

${\displaystyle {\overline {c}}_{m}(k,S)=\sum _{\stackrel {c_{0}+c_{1}+...+c_{m-1}=k}{c_{i}\in S\cap I_{k+1},i\in I_{m}}}{\frac {k!}{c_{0}!c_{1}!...c_{m-1}!}}\,}$

## Partition of a k-set over set S

${\displaystyle {\overline {p}}_{m}(k,S)=\sum _{\begin{matrix}\scriptstyle t_{0}+t_{1}+...+t_{k}=m\\\scriptstyle t_{1}+2t_{2}+...+kt_{k}=k\\\scriptstyle t_{i}=0,i\notin S\cap I_{k+1}\end{matrix}}{\frac {k!}{t_{1}!t_{2}!2!^{t_{2}}...t_{k}!k!^{t_{k}}}}\,}$