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?

Question

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?

graulhavav9 · Accepted Answer

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        .

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}}

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