Austin Roberts - cp's OEIS frontend

User: Austin Roberts

Austin Roberts has authored 2 sequences.

A195924 The number of fixed points in S_n by the action of Foata's bijection.

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

Offset: 0

Austin Roberts Oct 26 2011

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.

			Below are the orbits of S_4 in order of size. The first 10 are fixed points.
[(1, 2, 3, 4)]
[(2, 1, 3, 4)]
[(2, 3, 1, 4)]
[(2, 3, 4, 1)]
[(3, 2, 1, 4)]
[(3, 2, 4, 1)]
[(3, 4, 2, 1)]
[(4, 3, 2, 1)]
[(4, 1, 3, 2)]
[(1, 4, 2, 3)]
[(2, 4, 3, 1), (4, 2, 3, 1)]
[(1, 3, 2, 4), (3, 1, 2, 4)]
[(1, 4, 3, 2), (4, 3, 1, 2)]
[(1, 2, 4, 3), (4, 1, 2, 3)]
[(2, 1, 4, 3), (4, 2, 1, 3), (2, 4, 1, 3)]
[(1, 3, 4, 2), (3, 1, 4, 2), (3, 4, 1, 2)]

James Pfieffer, personal communication.

Dominique Foata and Marcel-Paul Schützenberger, Major Index and inversion number of permutations , Math. Nachr. 83 (1978), 143-159

Cf. A195924, A195931, A065161.

A195931 The number of orbits in S_n by the action of Foata's bijection.

Original entry on oeis.org

1, 1, 2, 5, 16, 56, 236, 998, 4544, 20346

Offset: 0

Austin Roberts, Oct 26 2011

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.

			The orbits of S_4 are:
[(1, 2, 3, 4)]
[(2, 1, 3, 4)]
[(2, 3, 1, 4)]
[(2, 3, 4, 1)]
[(3, 2, 1, 4)]
[(3, 2, 4, 1)]
[(3, 4, 2, 1)]
[(4, 3, 2, 1)]
[(2, 1, 4, 3), (4, 2, 1, 3), (2, 4, 1, 3)]
[(2, 4, 3, 1), (4, 2, 3, 1)]
[(1, 3, 2, 4), (3, 1, 2, 4)]
[(1, 3, 4, 2), (3, 1, 4, 2), (3, 4, 1, 2)]
[(1, 4, 3, 2), (4, 3, 1, 2)]
[(4, 1, 3, 2)]
[(1, 2, 4, 3), (4, 1, 2, 3)]
[(1, 4, 2, 3)]

James Pfeiffer, personal communication.

Dominique Foata and Marcel-Paul Schützenberger, Major Index and inversion number of permutations, Math. Nachr. 83 (1978), 143-159

Cf. A195931, A195924, A065161

cp's OEIS Frontend

User: Austin Roberts

A195924 The number of fixed points in S_n by the action of Foata's bijection.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs