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%I #37 Aug 05 2024 05:34:58
%S 1,2,1,1,3,1,2,1,1,2,1,1,1,4,1,2,1,1,3,1,2,1,1,2,1,1,1,3,1,2,1,1,2,1,
%T 1,1,2,1,1,1,1,5,1,2,1,1,3,1,2,1,1,2,1,1,1,4,1,2,1,1,3,1,2,1,1,2,1,1,
%U 1,3,1,2,1,1,2,1,1,1,2,1,1,1,1,4,1,2,1,1,3,1
%N a(n) is the position, counted from the right, of the rightmost nonzero value in the n-th nonzero restricted growth string in A239903 and its infinite continuation.
%C The integer a(n) intervenes as a parameter of an algorithm (Dejter) that generates an ordered tree comprising all Dyck words. The nodes of this ordered tree represent the cyclic classes of vertices, and the cycles of uniform 2-factors (Mutze et al.), of the Odd graphs and Middle-Levels graphs. These factors were used to generate explicit Hamilton cycles in those graphs.
%D I. J. Dejter, A numeral system for the middle-levels graph, EJGTA, Vol. 9 no. 1 (2021).
%D I. J. Dejter, Reinterpreting the middle-levels via natural enumeration of ordered trees, Vol. 3 (2020) Open Journal of Discrete Applied Mathematics, Vol 3 (2020) issue 2, pp. 8-22.
%D P. Gregor, T. Mutze, and J. Nummenpalo, A short proof of the middle-levels theorem, Discrete Analysis, 2018-8, 12 pp.
%D T. Mutze, C. Standke, and V. Wiechert, A minimum-change version of the Chung-Feller theorem for Dyck paths, European J. Combin, 69 (2018), 260-275.
%D T. Mutze, J. Nummenpalo, and D. Walczak, Sparse Kneser graphs are hamiltonian, J. London Math. Soc., 103 (2021), 1253-1275.
%D T. Mutze, Proof of the middle-levels conjecture, Proc. London Math. Soc., 112 (2016) 677-713.
%D J. Arndt, Matters Computational, Ideas, Algorithms, Source Code, Springer 2011 (fxtbook.pdf)
%H I. J. Dejter, <a href="http://arxiv.org/abs/1012.0995">A system of numeration for middle levels</a>, arXiv:1012.0995 [math.CO], (2014).
%H I. J. Dejter, <a href="https://www.researchgate.net/publication/245576352_On_a_lexical_tree_for_the_middle-levels_graph_problem">The role of restricted growth strings in the two middle levels of the Boolean lattice B_(2k+1)</a>, University of Puerto Rico, 2018.
%H I. J. Dejter, <a href="https://arxiv.org/abs/1911.02100">Reinterpreting Mütze's theorem via natural enumeration of ordered rooted trees</a>, arXiv:1911.02100 [math.CO], (2019).
%H I. J. Dejter, <a href="https://doi.org/10.5614/ejgta.2021.9.1.13">A numeral system for the middle-levels graphs</a>, Elec. J. Graph Theory and Applications (2021) Vol. 9, No. 1, 137-156. See p. 138.
%H I. J. Dejter, <a href="https://arxiv.org/abs/2203.05326">Edge-supplementary arc factorizations of odd graphs, uniform 2-factors and Hamilton cycles</a>, arXiv:2203.05326 [math.CO], 2022.
%H I. J. Dejter, <a href="https://arxiv.org/abs/2209.11122">Integer sequence of Dyck-path single changes</a>, arXiv:2209.11122 [math.CO], 2022.
%e The subindices j of the rightmost nonzero entries in the first five Catalan words a_{k-1}...a_j...a_1, namely a_1=1, a_2a_1=10, a_2a_1=11, a_2a_1=12, a_3a_2a_1=100, are 1,2,1,1,3.
%o (PARI) \\ here nxt(w) sequences reversed rgs words starting with [1].
%o nxt(w)=if(w[1]==#w, vector(#w+1, i, i>#w), my(k=1); while(w[k]>w[k+1], w[k]=0; k++); w[k]++; w)
%o seq(n)={my(a=vector(n), w=[1]); for(i=1, n, my(j=1); while(!w[j], j++); a[i]=j; w=nxt(w)); a} \\ _Andrew Howroyd_, Jan 23 2023
%Y Cf. A239903.
%K nonn,changed
%O 1,2
%A _Italo J Dejter_, Jan 23 2023
%E Terms a(43) and beyond from _Andrew Howroyd_, Jan 23 2023
%E Name corrected by _Italo J Dejter_, Feb 14 2023
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