How many ways can a permutation be factored into two n-cycles?

where (ral):i) denotes the permutation whfsh transposes natural embedding of S,, I in S,,. Note that

I

with nP and a,, Is the

P4i = Prb, = [ P(t5P9tZ)]7T,i ’ are not all distinct (and not just when i = n). The algebraic definitions of 41 and 4,, above are used in proving Lemma 2 and the basic recursion on f(P) given in Theorem 2. Lemma 2 may be viewed as a much expanded enumerative version of Gleason’s existence proposition [3]. Lemma 1 gkes an alternate graphical kterpretation of & and 4, in arms of the cycle structures of the permutations involved. dhenever (n, i, UP)

Theorem 2. f(P)=C,+i+,pf(P&).

Lemma 1. The cycle slructure of P&i may be obtained from P by one of the fclfowin~:

Pro&~, Theorem 2 is an elementary esn~equenee of (e) and (h) sf Lemma 2, Lemma 1 1~ eadly cheeked by ~isudib~g I) and P+i as dirge&d graphs, In particular, I%#,is obtained from B in Ic) and (d) by dsleting it and it8 incident arcs, adding an arc Pram 1119’ to nP, and then switchingthe destination of this arc with that of the are from 18. I to I, Part (d af Lemma 2 is obvious, (The terms far I - n and I = nP omitted Pram the formula far F(Q) carreiipand to impassible csnditianfi nV = n and /ICIg n,) TO ~IQV~J if) it SU#~XS to ~‘ht~k first that Cpiand Cpfida indeed Send S,f,iand St! 1 into each other and then verify algebraically that their composition in either order is an identity, Part (g) is derived from (4)by verifying (using reasoning as in (b)) that & and +$*isend C’,l,iand C’,,-1into e@chother, The proof 04 (h) is similar. Thus, the computation

(U&,(V~,,)= [ Uh nwh

nw17G'

= UV(nV, nUV)(n, nVh;’ = wJW4,” together with (g) shaws Qi: F(P)

+

F(f+i)e

Similarly,

= (U’V’)W&, i)(i, nP) = (U’V’)t#i~n, shows @$+ : F(P4i) + F,(P). Part (g) shows directly that three of the four compositions of the components of @ and @f,,p are identities. The fourth identity, = k. W&Rk = U(nU, n)(n, k) = U, follows from nU = nPV_’ ==nP(V&&J

(n-3)ff(n-1)+(at,-~)~n-3)f’(n-3), (n-4)f’(n-1)~2(n-2)f’(ra-2)),

(2)

where i ’ in 1 is the maximum of f(P) for P in A, n QI, The three caaea in (2) reflcet three possible values for I& P)!, namely, 2, 3, sn8 ~4, The extrs term@ arc introduced via (1, for the same regllontl given at the end of Section 2. A tcdbus expendan of polynomials of degree at morJt 6 will lahow that n-2

h’02b=W-2)!

n

(l+2/k2)