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A178446 Number of perfect matchings in the Triangle Graph of order n, reduced by the spire vertex if n mod 4 equals 1 or 2. 4
1, 1, 1, 2, 6, 28, 200, 2196, 37004, 957304, 38016960, 2317631400, 216893681800, 31159166587056, 6871649018572800, 2326335506123418128, 1208982377794384163088, 964503557426086478029152, 1181201363574177619007442944, 2220650888749669503773432361504, 6408743336016148761893699822360672 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

The Triangle Graph of order n has n rows with i vertices in row i. Each vertex is connected to the neighbors in the same row and up to two vertices in each of the neighboring rows. The Graph has A000217(n) vertices and 3*A000217(n-1) edges altogether.

In order to be able to find matchings the Triangle Graph of order n is reduced by the spire vertex (one vertex in row 1) and the incident edges if n mod 4 is in {1, 2}.  The resulting graph has an even number of vertices.

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..60

Propp, J., Enumeration of matchings: problems and progress, in L. J. Billera et al. (eds.), New Perspectives in Algebraic Combinatorics

Eric Weisstein's World of Mathematics, Perfect Matching

Eric Weisstein's World of Mathematics, Triangle Graph

EXAMPLE

4 example graphs:                       o

.                                      / \

.                          o          o---o

.                         / \        / \ / \

.                ( )     o---o      o---o---o

.                       / \ / \    / \ / \ / \

.           ( ) o---o  o---o---o  o---o---o---o

Order:       1    2        3            4

Vertices:    0    2        6           10

Edges:       0    1        9           18

Matchings:   1    1        2            6

MAPLE

with(LinearAlgebra):

a:= proc(n) option remember; local i, j, h0, h1, M, s, t;

      if n<2 then 1

    else s:= `if`(member(irem(n, 4), [1, 2]), 1, 0);

         M:= Matrix((n+1)*n/2 -s, shape=skewsymmetric);

         if s=1 then M[1, 2]:=1 fi;

         for j from 1+s to n-1 do

           h0:= j*(j-1)/2 +1-s;

           h1:= h0+j;

           t:= 1;

           for i from 1 to j do

             M[h1, h1+1]:= 1;

             M[h1, h0]:= t;

             h1:= h1+1;

             M[h1, h0]:= t;

             h0:= h0+1;

             t:= -t

           od

         od;

         sqrt(Determinant(M))

      fi

    end:

seq(a(n), n=0..15);

CROSSREFS

Cf. A039907, A071093, A182797.

Sequence in context: A229112 A201959 A216187 * A324126 A272662 A125812

Adjacent sequences:  A178443 A178444 A178445 * A178447 A178448 A178449

KEYWORD

nonn

AUTHOR

Alois P. Heinz, Dec 24 2010

STATUS

approved

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Last modified June 6 15:48 EDT 2020. Contains 334828 sequences. (Running on oeis4.)