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A346198
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a(n) is the number of permutations on [n] with no strong fixed points but contains at least one small descent.
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3
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0, 1, 1, 8, 43, 283, 2126, 17947, 168461, 1741824, 19684171, 241506539, 3198239994, 45482655683, 691471698917, 11193266251700, 192238116358427, 3491633681792507, 66875708261486766, 1347168876070616179, 28474546456352896021, 630130731702950549248, 14570725407559756078387, 351411668456841530417027
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OFFSET
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1,4
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COMMENTS
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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.
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REFERENCES
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E. R. Berlekamp, J. H. Conway, and R. K. Guy, Winning Ways For Your Mathematical Plays, Vol. 1, CRC Press, 2001.
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LINKS
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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.
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FORMULA
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For n > 2, a(n) = b(n)-c(n) where b(n) = A052186(n-1), c(n) = A346189(n).
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EXAMPLE
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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.
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PROG
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(Python) See A346204.
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CROSSREFS
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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
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KEYWORD
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nonn
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AUTHOR
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Eugene Fiorini, Jared Glassband, Garrison Lee Koch, Sophia Lebiere, Xufei Liu, Evan Sabini, Nathan B. Shank, Andrew Woldar, Jul 09 2021
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STATUS
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approved
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