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A333348 Matching number of the tree of n vertices with the largest number of maximum matchings. 1
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, 10, 10, 10, 10, 11, 11, 11, 12, 12, 12, 12, 13, 13, 13, 14, 14, 14, 14, 15, 15, 15, 16, 16, 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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,8

COMMENTS

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.

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.

LINKS

Table of n, a(n) for n=0..85.

Clemens Heuberger and Stephan Wagner, The Number of Maximum Matchings in a Tree, Discrete Mathematics, volume 311, issue 21, November 2011, pages 2512-2542; arXiv preprint, arXiv:1011.6554 [math.CO], 2010.

Clemens Heuberger and Stephan Wagner, Number of Maximum Matchings In a Tree - Sage Worksheet, constructing the trees.

Kevin Ryde, vpar examples/most-maximum-matchings.gp creating, counting, and recurrences, in Pari/GP.

Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,1,-1).

FORMULA

a(2)=a(6)=1, a(13)=3, a(20)=5, and otherwise a(n) = floor((2n+2)/7).

CROSSREFS

Cf. A333347 (number of maximum matchings).

Sequence in context: A051889 A086707 A217538 * A194920 A327440 A285588

Adjacent sequences:  A333345 A333346 A333347 * A333349 A333350 A333351

KEYWORD

nonn

AUTHOR

Kevin Ryde, Mar 15 2020

STATUS

approved

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Last modified September 29 10:03 EDT 2020. Contains 337428 sequences. (Running on oeis4.)