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A126301 A071158-codes for the fixed points of Vaillé's 1997 bijection on Dyck paths.

%I A126301 #8 Mar 22 2016 06:07:05
%S A126301 0,1,23211,2432211,2354543212221,335465432122211
%N A126301 A071158-codes for the fixed points of Vaillé's 1997 bijection on Dyck paths.
%C A126301 Vaille gives the terms a(2)-a(4) on the last page of the 1997 paper. Note that this sequence migh be finite, for two reasons: (a) there are no more fixed points after some limit (the next one after a(5) must have at least 19 digits. All the terms must be of odd length). (b) some of the fixed points would require digits higher than "9", in which case the factorial expansion can nomore presented unambiguously in decimal. However, the sequence A126311 can accommodate also those cases.
%H A126301 J. Vaillé, <a href="http://dx.doi.org/10.1006/eujc.1996.0089">Une Bijection Explicative de Plusieurs Propriétés Remarquables des Ponts</a>, European J. Combin. 18 (1997), no. 1, 117-124.
%e A126301 This sequence consists of those terms of A071158 for which the first factorial digit is equal to the number of 1's in the term and the following algorithm results the remaining factorial digits of the same term: First, extract all maximal subsequences from the term (for d ranging from 1 to the largest digit present) that consist of digits d and d+1 and place them next to each other, from left to right. E.g. for the term 2354543212221 this yields the sequence: 2212221,2332222,3443,5454,55. After discarding the last digit (here 5) and replacing in each batch the smaller number with +1 and larger number with -1, we get:
%e A126301 -1,-1,+1,-1,-1,-1,+1,+1,-1,-1,+1,+1,+1,+1,+1,-1,-1,+1,-1,+1,-1,+1,+1.
%e A126301 and summing these from RIGHT, we get the following partial sums:
%e A126301 1, 2, 3, 2, 3, 4, 5, 4, 3, 4, 5, 4, 3, 2, 1, 0, 1, 2, 1, 2, 1, 2, 1.
%e A126301 Retaining only the partial sums under the +1's (i.e. the rightmost one and all the partial sums that are larger than the preceding partial sum one step to the right) we obtain: 3,5,4,5,4,3,2,1,2,2,2 and 1. These, after appended to the number of 1's in the original term (2), yields the same term 2354543212221 from which we started from, which thus is a member of this sequence. Similarly, the term 2432211 belongs to this sequence, because the same procedure yields:
%e A126301 22211,2322,43,4 and after discarding the last 4:
%e A126301 -1,-1,-1,+1,+1,+1,-1,+1,+1,-1,+1 and summing from the right:
%e A126301 1, 2, 3, 4, 3, 2, 1, 2, 1, 0, 1.
%e A126301 collecting all the partial sums larger than their right neighbor (those under +1's), which appended after the number of 1's (2), results the same term 2432211.
%Y A126301 a(n) = A071158(A126300(n)) = A007623(A126311(n)). Subset of A126299. Cf. A126295.
%K A126301 nonn,hard,more,base
%O A126301 0,3
%A A126301 _Antti Karttunen_, Jan 02 2007