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

Author

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

Keywords

• nonn

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.

Examples

			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.
		

References

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

Links

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

Crossrefs

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

Programs

Python

See A346204.

Formula

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