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.
- Restricted compositions of natural numbers
- Generalized Pascal triangle
- Compositions of_natural numbers over arithmetic progressions
- Compositions and Partitions of sets over Np
Note on notations
- (1)......
the set of natural numbers
- (2)......
![{\displaystyle I_{a}^{b}=\{x:b\leq x<a,x\in N\}\,}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/cf0f9ee8ca1b5c2b17252e7f264d15bab8f03e88)
- (3)......
![{\displaystyle I_{a}^{0}=I_{a}\,}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/215f375cfdaaafa5c5f1ce89453eb733523ad3e7)
- (4)......
![{\displaystyle I_{a+1}^{1}=N_{a}\,}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/e801b8fa35352b2e73d63542af5e23784563b1cf)
- (5)......
![{\displaystyle I_{\infty }^{b}=I^{b}\,}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/8fae6ed273f8de3fe060a589507e39f9052720f4)
- (6)......
![{\displaystyle O=\{x:x=2n+1,n\in N\}\,}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/2f0a24cc2d51863a054643b3b23ff5ab0156d2e1)
- (7)......
![{\displaystyle E=\{x:x=2n,n\in N\}\,}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/6d3157350e40ba746996a885ecb758bb38f8a3f0)
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}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/196e450710c0fd411f41af574f52eb055c4135f4)
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\,}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/b08b9f90545225db5241bdd1db98a4f4441e8ff6)
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}!}}\,}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/d7ea4c257dc408ab33c6447a1955d4495a2ae04e)
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}}}}\,}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/43a96d4fd78a623b63b1f540282c389d8fa7c28b)