Prove that number of partitions of a k-set into m parts of sizes, from certain set S that is subset of {0,1,2,…} is P_{m}(k,S)=\sum_{\sum_{s\in S}t_s=m,\sum_{s\in S}st_s=k}\prod_{s\in S-{0}}\frac{k!}{t_s!(s!)^{t_s}}

jhenezhubby01ff 2022-10-02 Answered
Prove that number of partitions of a k-set into m parts of sizes, from certain set S that is subset of { 0 , 1 , 2 , } is
P m ( k , S ) = s S t s = m , s S s t s = k s S 0 k ! t s ! ( s ! ) t s
and generating function considering parameter k is
k = 0 P m ( k , S ) x k k ! = t 0 = 0 m 1 1 ( m t 0 ) ! ( s S 0 x s s ! ) m t 0
Can someone find the generating function with two variables considering parameters m and k?
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Answers (1)

graulhavav9
Answered 2022-10-03 Author has 14 answers
One wants to compute the two variables generating function
F S ( x , y ) = m = 1 y m k = 0 P m ( k , S ) x k k ! .
Note that, according to the second displayed formula in your post,
k = 0 P m ( k , S ) x k k ! = i = 1 m 1 i ! u S ( x ) i , with   u S ( x ) = s S 0 x s s ! .
Hence,
F S ( x , y ) = i = 1 1 i ! u S ( x ) i m = i y m = i = 1 1 i ! u S ( x ) i y i 1 y ,
that is,
F S ( x , y ) = exp ( y u S ( x ) ) 1 1 y .
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