User:Adi Dani

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.

1. Restricted compositions of natural numbers
2. Generalized Pascal triangle
3. Compositions of_natural numbers over arithmetic progressions
4. Compositions and Partitions of sets over N_p

Note on notations

(1)......$N=\{0,1,2,...\}\,$ the set of natural numbers

(2)......$I_{a}^{b}=\{x:b\leq x<a,x\in N\}\,$

(3)......$I_{a}^{0}=I_{a}\,$

(4)......$I_{a+1}^{1}=N_{a}\,$

(5)......$I_{\infty }^{b}=I^{b}\,$

(6)......$O=\{x:x=2n+1,n\in N\}\,$

(7)......$E=\{x:x=2n,n\in N\}\,$

Composition of natural number k over set S

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

Partitions of natural number k over set S

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

Composition of a k-set over set S

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

Partition of a k-set over set S

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