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%I #47 Aug 07 2021 04:15:00
%S 0,1,1,8,43,283,2126,17947,168461,1741824,19684171,241506539,
%T 3198239994,45482655683,691471698917,11193266251700,192238116358427,
%U 3491633681792507,66875708261486766,1347168876070616179,28474546456352896021,630130731702950549248,14570725407559756078387,351411668456841530417027
%N a(n) is the number of permutations on [n] with no strong fixed points but contains at least one small descent.
%C A small descent in a permutation p is a position i such that p(i)-p(i+1)=1.
%C 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.
%D E. R. Berlekamp, J. H. Conway, and R. K. Guy, Winning Ways For Your Mathematical Plays, Vol. 1, CRC Press, 2001.
%H M. Lind, E. Fiorini, A. Woldar, and W. H. T. Wong, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL24/Wong/wong31.html">On Properties of Pebble Assignment Graphs</a>, Journal of Integer Sequences, 24(6), 2020.
%F For n > 2, a(n) = b(n)-c(n) where b(n) = A052186(n-1), c(n) = A346189(n).
%e 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.
%o (Python) See A346204.
%Y Cf. A000255, A000166, A000153, A000261, A001909, A001910, A055790, A346189, A346199, A346204.
%K nonn
%O 1,4
%A _Eugene Fiorini_, _Jared Glassband_, _Garrison Lee Koch_, _Sophia Lebiere_, _Xufei Liu_, _Evan Sabini_, _Nathan B. Shank_, _Andrew Woldar_, Jul 09 2021