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%I #56 Aug 22 2021 18:52:10
%S 1,1,1,5,19,95,569,3957,31455,281435,2799981,30666153,366646995,
%T 4751669391,66348304849,992975080813,15856445382119,269096399032035,
%U 4836375742967861,91766664243841393,1833100630242606203,38452789552631651191,845116020421125048153
%N a(n) is the number of permutations on [n] with at least one strong fixed point and no small descents.
%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 a(n) = b(n-1) + Sum_{i=4..n} A346189(i-1)*b(n-i) where b(n) = A000255(n).
%e For n = 4, the a(4) = 5 permutations on [4] with strong fixed points but no small descents: {(1*, 2*, 3*, 4*), (1*, 3, 4, 2), (1*, 4, 2, 3), (2, 3, 1, 4*), (3, 1, 2, 4*)} where * marks strong fixed points.
%o (Python) See A346204.
%Y Cf. A000255, A000166, A000153, A000261, A001909, A001910, A055790, A346189, A346198, 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