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%I #27 Aug 22 2021 13:28:33
%S 0,0,1,1,1,1,1,2,2,2,3,3,3,3,4,4,4,5,5,5,5,6,6,6,7,7,7,8,8,8,8,9,9,9,
%T 10,10,10,10,11,11,11,12,12,12,12,13,13,13,14,14,14,14,15,15,15,16,16,
%U 16,16,17,17,17,18,18,18,18,19,19,19,20,20,20,20,21,21,21,22,22,22,22,23,23,23,24,24,24
%N Matching number of the tree of n vertices with the largest number of maximum matchings.
%C Heuberger and Wagner consider how many different maximum matchings a tree of n vertices may have. They determine the unique tree (free tree) of n vertices with the largest number of maximum matchings, or at n=6 and n=34 the two trees with equal largest number. a(n) is the matching number of the unique tree, and of both n=34 trees since they have the same matching number. For n=6, a(6)=1 is the star-6 which is their T_{6,1}. The other n=6 is their T_{6,2} and its matching number would be a(6)=2 instead.
%C The trees n!=2 have all pairs of leaves an even distance apart (the type of free tree counted by A304867). Vertices an even distance to a leaf are Heuberger and Wagner's type A, and vertices an odd distance to a leaf are type B. Per their definitions (and for any "even distance leaves" tree in fact), all type B vertices must be matched in a maximum matching and consequently the matching number is the number of type B vertices. 2n/7 appears in the formula below since each "C" part contains 7 vertices of which 2 are type B; then there are certain fixed additional B vertices according to n mod 7.
%H Clemens Heuberger and Stephan Wagner, <a href="https://doi.org/10.1016/j.disc.2011.07.028">The Number of Maximum Matchings in a Tree</a>, Discrete Mathematics, volume 311, issue 21, November 2011, pages 2512-2542; <a href="https://arxiv.org/abs/1011.6554">arXiv preprint</a>, arXiv:1011.6554 [math.CO], 2010.
%H Clemens Heuberger and Stephan Wagner, <a href="https://www.math.tugraz.at/~cheub/publications/max-card-matching/">Number of Maximum Matchings In a Tree - Sage Worksheet</a>, constructing the trees.
%H Kevin Ryde, <a href="http://user42.tuxfamily.org/pari-vpar/index.html">vpar</a> examples/most-maximum-matchings.gp creating, counting, and recurrences, in PARI/GP.
%H <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,0,0,0,1,-1).
%F a(2)=a(6)=1, a(13)=3, a(20)=5, and otherwise a(n) = floor((2n+2)/7).
%Y Cf. A333347 (number of maximum matchings).
%K nonn
%O 0,8
%A _Kevin Ryde_, Mar 15 2020
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