A195924 - OEIS

%I #22 Jan 08 2013 18:56:37

%S 1,1,2,4,10,26,80,256,918,3464

%N The number of fixed points in S_n by the action of Foata's bijection.

%C Foata's bijection takes a permutation w with maj(w) = x to a permutation F(w) with inv(F(w)) = x. Applying F repeatedly partitions the symmetric group into distinct orbits. F also preserves inverse descent sets.

%D James Pfieffer, personal communication.

%H Dominique Foata and Marcel-Paul Schützenberger, <a href="http://dx.doi.org/10.1002/mana.19780830111">Major Index and inversion number of permutations </a>, Math. Nachr. 83 (1978), 143-159

%e Below are the orbits of S_4 in order of size. The first 10 are fixed points.

%e [(1, 2, 3, 4)]

%e [(2, 1, 3, 4)]

%e [(2, 3, 1, 4)]

%e [(2, 3, 4, 1)]

%e [(3, 2, 1, 4)]

%e [(3, 2, 4, 1)]

%e [(3, 4, 2, 1)]

%e [(4, 3, 2, 1)]

%e [(4, 1, 3, 2)]

%e [(1, 4, 2, 3)]

%e [(2, 4, 3, 1), (4, 2, 3, 1)]

%e [(1, 3, 2, 4), (3, 1, 2, 4)]

%e [(1, 4, 3, 2), (4, 3, 1, 2)]

%e [(1, 2, 4, 3), (4, 1, 2, 3)]

%e [(2, 1, 4, 3), (4, 2, 1, 3), (2, 4, 1, 3)]

%e [(1, 3, 4, 2), (3, 1, 4, 2), (3, 4, 1, 2)]

%Y Cf. A195924, A195931, A065161.

%K nonn,hard,more

%O 0,3

%A _Austin Roberts_ Oct 26 2011