Ndryshimi mes inspektimeve të "Përdoruesi:Armend/nënfaqe"

is called composition of natural number k in m parts over <math>A_{s}^{p}\,</math>
<br>Denote by <math>C_{m}(k,A_{s}^{p})\,</math> the set of compositions of natural number k in m parts over <math>A_{s}^{p}\,</math>
== generating function &nbsp; &nbsp; ==
 
<br>
 
&nbsp; &nbsp; &nbsp;&nbsp;<math>\sum_{k=0}^{\infty}c_{m}(k,A_{s}^{0}-\{0\})x^{k}=\sum_{c_{i}\in A_{s}^{p}-\{0\},i\in I_{m}}x^{c_{0}+c_{1}+...+c_{m-1}}=\,</math>
 
<br> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<math>=\sum_{c_{0}\in A_{s}^{0}-\{0\}}x^{c_{0}}\sum_{c_{1}\in A_{s}^{0}-\{0\}}x^{c_{1}}...\sum_{c_{m-1}\in A_{s}^{0}-\{0\}}x^{c_{m-1}}=\,</math>
 
<br> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<math>=\left(\sum_{i=1}^{\infty}x^{si}\right)^{m}=\left(\frac{x^{s+1}}{1-x} \right)^{m}\,</math>
 
<br> now from binomial formula we get
 
<br> &nbsp; &nbsp; &nbsp; &nbsp;<math>\left(\frac{x^{p}}{1-x^{s}}\right)^{m}=x^{pm}(x^{s})^{-m}=x^{pm}\sum_{i=0}^{\infty}\binom{-m}{i}(-x^{s})^{i}=\,</math>
 
<br> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp;<math>=\sum_{i=0}^{\infty}\binom{m+i-1}{i}x^{pm+si}</math>
 
<br> if now substitute <math>pm+si=k\,</math>&nbsp;&nbsp;thent &nbsp;<math>i=(k-pm)/s\,</math>&nbsp; taking in account that <math>i\in N\,</math> follow
 
that <math>k\in\{pm,pm+s,pm+2s,...\},</math>