The asymtotics of the number of permutations pi of [1,n] such that 2 pi = IdentityPerm up to order, 5, is equal to (n/2) 1/2 / 7 119 0.550695314903185 exp(- n/2) n exp(n ) |1 + ------- - ------ | 1/2 1152 n \ 24 n 7933 1967381 57200419 562799 - ----------- + ----------- - --------------- - ----------- 3/2 2 5/2 3 414720 n 39813120 n 1337720832 n 47775744 n 526420847 1856209 267645803 \ - ---------------- + ------------ - ------------------| 7/2 4 9/2| 40131624960 n 573308928 n 2407897497600 n / which in floating-point is (0.5000000000 n) 1/2 / 0.5506953149 exp(-0.5000000000 n) n exp(n ) |1. | \ 0.2916666667 0.1032986111 0.01912856867 0.04941539372 + ------------ - ------------ - ------------- + ------------- 1/2 n 3/2 2 n n n 0.04275960846 0.01178001540 0.01311735689 0.003237711658 - ------------- - ------------- - ------------- + -------------- 5/2 3 7/2 4 n n n n 0.0001111533208\ - ---------------| 9/2 | n / The asymtotics of the number of permutations pi of [1,n] such that 3 pi = IdentityPerm up to order, 5, is equal to /2 n\ |---| 2 n \ 3 / (1/3) / 1 25 289 0.5773504 exp(- ---) n exp(n ) |1 - -------- + ------- - ------ 3 | (1/3) 2/3 1296 n \ 6 n 72 n 1 25 613 1 25 11381 - --------- + -------- - ------- - --------- + -------- - -------- (4/3) 5/3 2 (7/3) 8/3 3 12 n 144 n 2592 n 48 n 576 n 51840 n 31 155 6529 \ + ---------- - ---------- + ---------| 10/3 11/3 4| 1440 n 3456 n 311040 n / which in floating-point is (0.6666666667 n) (1/3) / 0.1666666667 0.5773504 exp(-0.6666666667 n) n exp(n ) |1. - ------------ | 1/3 \ n 0.3472222222 0.2229938272 0.08333333333 0.1736111111 + ------------ - ------------ - ------------- + ------------ 2/3 n 4/3 5/3 n n n 0.2364969136 0.02083333333 0.04340277778 0.2195408951 - ------------ - ------------- + ------------- - ------------ 2 7/3 8/3 3 n n n n 0.02152777778 0.04484953704 0.02099086934\ + ------------- - ------------- + -------------| 10/3 11/3 4 | n n n / The asymtotics of the number of permutations pi of [1,n] such that 4 pi = IdentityPerm up to order, 5, is equal to /3 n \ |--- + 3/2| -6 3 n \ 4 / 0.1529440 10 exp(- ---) n 4 1/2 1/2 2 (3/4) (1/4) 4 n exp(1/4 RootOf(_Z - 2) 4 n + ---------) 4 / 1 11 59 1831 \ |1 + 1/n + ---- - ----- - ------ + ------| | 2 3 4 5| \ 2 n 30 n 120 n 840 n / which in floating-point is -6 (0.7500000000 n + 1.500000000) 0.1529440 10 exp(-0.7500000000 n) n (1/4) 1/2 exp(0.9999999998 n + 0.5000000000 n ) / 0.5000000000 0.3666666667 0.4916666667 2.179761905\ |1. + 1/n + ------------ - ------------ - ------------ + -----------| | 2 3 4 5 | \ n n n n / The asymtotics of the number of permutations pi of [1,n] such that 5 pi = IdentityPerm up to order, 5, is equal to /4 n \ |--- + 2| -8 4 n \ 5 / (1/5) 0.1130041 10 exp(- ---) n exp(n ) 5 / 5 25 50 3875 330625 \ |1 + --- + ----- - ----- - ------- + --------| | 3 n 2 3 4 5| \ 18 n 81 n 1944 n 40824 n / which in floating-point is -8 (0.8000000000 n + 2.) (1/5) 0.1130041 10 exp(-0.8000000000 n) n exp(n ) / 1.666666667 1.388888889 0.6172839506 1.993312757 8.098789927\ |1. + ----------- + ----------- - ------------ - ----------- + -----------| | n 2 3 4 5 | \ n n n n / The asymtotics of the number of permutations pi of [1,n] such that 6 pi = IdentityPerm up to order, 5, is equal to /5 n \ |--- + 5/2| 5 n \ 6 / 0.000916 exp(- ---) n 6 2 1/2 1/2 exp(1/6 RootOf(-2 + 3 _Z , label = _L19) 6 n ) / 5 25 19 2255 119863 \ |1 + --- + ---- - ----- - ------ + -------| | 2 n 2 3 4 5| \ 8 n 48 n 384 n 5376 n / which in floating-point is (0.8333333333 n + 2.500000000) 0.000916 exp(-0.8333333333 n) n 1/2 exp(0.3333333334 n ) / 2.500000000 3.125000000 0.3958333333 5.872395833 22.29594494\ |1. + ----------- + ----------- - ------------ - ----------- + -----------| | n 2 3 4 5 | \ n n n n / Everything is rigorous, but the constants in front are non-rigorous (yet fairly reliable) estimates This took, 47.948, seconds of CPU time