Question

find a closed formula for c(n,n-4), for all n>=4

use theorem below

x G ueth yple type ( )

Answer 1

closed formula for c(n. 0-4) o breave that there cannot be a cycle of length at least 6 because that leaves n-6 items! which can form at most n-6 cycles for a total of n. 5 Eycles. Similarly for a cycle of length seven and so on. S o The species of perustations coith geas of length at most five is PCAIC= 1 (2) + A26= 22) +- + A 50 = 52)) This gives the generating function G(z) = expl Zag 22 ) 229 Now there are several cases. first case A five cycle and 3.5 fixed points. This gives the generating fuxtion Ejns bap (0, 2)expla; 3) exp (ar 2) we can doop, the contributions in A2, A, at and no such cycles appear because there aren't any items left over, for an arswer of 2-1 251 che); (3) 39) second case. A four cycle, a two cycle and

n6 fined points for a contribution of ( ) ( ) Third case Two thau-cycles and nos fixed points, for a contribution of (26)! (7) 0 - ( 2 ) Fourth care A three cycle, two two-cycles and no7 fixed points for a contribution of ce) (3) 24 (2) Adding the contributions from these generating functions we obtain 2 + + + + (1-5)(n-1) + (m-5) (n-6) (6-72) z . In + Inti (25) (384 192 1152 2880 Performing coefficient extraction on this we obtain the answer (n.5)1 384 192 1152 2280 www. (n-5)/ ALYVORE 384 1880