A346198 a(n) is the number of permutations on [n] with no strong fixed points but contains at least one small descent.

0, 1, 1, 8, 43, 283, 2126, 17947, 168461, 1741824, 19684171, 241506539, 3198239994, 45482655683, 691471698917, 11193266251700, 192238116358427, 3491633681792507, 66875708261486766, 1347168876070616179, 28474546456352896021, 630130731702950549248, 14570725407559756078387, 351411668456841530417027

Offset

1,4

Comments

A small descent in a permutation p is a position i such that p(i)-p(i+1)=1.

A strong fixed point is a fixed point (or splitter) p(k)=k such that p(i) < k for i < k and p(j) > k for j > k.

References

E. R. Berlekamp, J. H. Conway, and R. K. Guy, Winning Ways For Your Mathematical Plays, Vol. 1, CRC Press, 2001.

Links

Table of n, a(n) for n=1..24.

M. Lind, E. Fiorini, A. Woldar, and W. H. T. Wong, On Properties of Pebble Assignment Graphs, Journal of Integer Sequences, 24(6), 2020.

Formula

For n > 2, a(n) = b(n)-c(n) where b(n) = A052186(n-1), c(n) = A346189(n).

Example

For n = 4, the a(4) = 8 permutations on [4] with no strong fixed points but has small descents: {([2, 1], [4, 3]), (2, [4, 3], 1), ([3, 2], 4, 1), (3, 4, [2, 1]), (4, 1, [3, 2]), (4, [2, 1], 3), ([4, 3], 1, 2), (<4, 3, 2, 1>)} []small descent, <>consecutive small descents.

Prog

(Python) See A346204.

Crossrefs

Cf. A000255, A000166, A000153, A000261, A001909, A001910, A055790, A346189, A346199, A346204.

Sequence in context: A117617 A111365 A199321 * A144039 A282189 A210125

Adjacent sequences: A346195 A346196 A346197 * A346199 A346200 A346201

Keyword

nonn

Author

Eugene Fiorini, Jared Glassband, Garrison Lee Koch, Sophia Lebiere, Xufei Liu, Evan Sabini, Nathan B. Shank, Andrew Woldar, Jul 09 2021

Status

approved