限定条件下的有序多子集组计数问题

Counting Problems on the Ordered Multi-Subset Group under the Limited Conditions

研究了在限定条件下的有序多子集组计数问题,推导了在条件:(A_1∪…∪A_p)∪(B_1∩…∩B_q)=N_n,A_1,…,A_p,B_1,…,B_q■N_n下,集函数x_1^(|A_p|)…x_p^(|A_p|)y_1^(|B_1|)…y_q^(|B_q|)的相关计数式,得到了一个重要的定理:W_(n;p,q)(x,y)=Σ_A1,…,A_p,B_1,…,B_q■N_n(A_1∪…∪A_p)∪(B_1∩…∩B_q)=N_nx_1^(|A_p|)…x_p^(|A_p|)y_1^(|B_1|)…y_q^(|B_q|)={[f(X)-1]g(Y)+y_1y_2…yq}~n,其中f(X)=(1+x_1)(1+x_2)…(1+x_p),g(y)=(1+y_1)(1+y_2)…(1+y_q).并在此基础上,做了一系列推广及应用.

Counting problems on the ordered multi-subset group under the limited conditionsare studied. The counting formula of the set function＂x1^（｜Ap｜）…xp^（｜Ap｜）y1^（｜B1｜）…yq^（｜Bq｜）＂under the conditions are derived：（A1∪…∪Ap）∪（B1∩…∩Bq）= Nn,A1,…,Ap,B1,…,Bq∈Nn,and animportant theorem is obtained：W（n;p,q）（x,y）=ΣA1,…,Ap,B1,…,Bq∈Nn（A1∪…∪Ap）∪（B1∩…∩Bq）=Nnx1^（｜Ap｜）…xp^（｜Ap｜）y1^（｜B1｜）…yq^（｜Bq｜）={[f（X）-1]g（Y）＋y1y2…yq}^nAmong them,f（X）=（1＋x1）（1＋x2）…（1＋xp）,g（y）=（1＋y1）（1＋y2）…（1＋yq）There are a series of promotion and application based on the theorem.