有限集S上变换的标准分解及其应用

The standard factorization of transformation σ of S with applications

通过研究集合S={ 1,2 ,… ,n}上变换σ的动力系统性质 :(1)得到了σ的标准分解式 :σ =(q11q12 …q1t1 ) ∧ (q2 1q2 2 …q2t2 ) ∧ … (qm1qm2 …qmtm) ∧ (j11j12 …j1n1 ) (j2 1j2 2 …j2n2 )… (jk1jk2 …jknk) ;(2 )证明了 :｜H n ｜ =∑ni =0Cin(n-i) i(n-i) ! ,其中H n ={σ∈Hn｜σk+ 1=σ ,k =1,2 ,3,… } .

Let S be {1,2,3,:,n}, σ be a transformation of S. In this paper,we used dynamical systems ideals to prove the following results:;(1) The standard factorization of $σ:σ=(q-{11}q-{12}:q-{1t-1})-∧(q-{21}q-{22}:q-{2t-2})-∧:(q-{m1}q-{m2}:q-{mt-m})-∧(j-{11}j-{12}:j-{1n-1})(j-{21}j-{22}:j-{2n-2}):(j-{k1}j-{k2}:j-{kn-k})$;where q-{11},q-{21},:,q-{m1} are all primitive elements of S, q-{ip}=σ+{p-1}(q-{i1}) for p=1,2,:,t-i, i=1,2,:,m; (j-{i1}j-{i2}:j-{in-i}) is a n-i-cycle for i=1,2,:,k.;(2) |H+*-n|=∑ni=0C+i-n(n-i)+i(n-i)!, where H+*-n={σ∈H-n|σ+{k+1}=σ,k=1,2,3,:}, H-n=M(S).;