A071093 Number of perfect matchings in triangle graph with n nodes per side as n runs through numbers congruent to 0 or 3 mod 4.

1, 2, 6, 2196, 37004, 2317631400, 216893681800, 2326335506123418128, 1208982377794384163088, 2220650888749669503773432361504, 6408743336016148761893699822360672, 2015895925780490675949731718780144934779733312, 32307672245407537492814937397129549558917000333504

Offset

0,2

References

James Propp, Enumeration of matchings: problems and progress, pp. 255-291 in L. J. Billera et al., eds, New Perspectives in Algebraic Combinatorics, Cambridge, 1999 (see Problem 17).

Links

Alois P. Heinz, Table of n, a(n) for n = 0..44

James Propp, Twenty open problems in enumeration of matchings, arXiv:math/9801061 [math.CO], 1998-1999.

James Propp, Updated article

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

James Propp, Some 2-Adic Conjectures Concerning Polyomino Tilings of Aztec Diamonds, Integers (2023) Vol. 23, Art. A30.

Formula

a(2n) = A039907(4n) = A178446(4n), a(2n+1) = A039907(4n+3) = A178446(4n+3). - Andrew Howroyd, Mar 06 2016

Maple

with(LinearAlgebra): b:= proc(n) option remember; local l, ll, i, j, h0, h1, M; if n=0 then return 1 fi; if n<0 or member(irem(n, 4), [1, 2]) then return 0 fi; l:= []; for j from 1 to n-1 do h0:= j*(j-1)/2+1; h1:= j*(j+1)/2+1; for i from 1 to j do l:= [l[], [h1, h1+1]]; if irem(i, 2)=1 then l:= [l[], [h1, h0]]; h1:= h1+1; l:=[l[], [h1, h0]]; h0:=h0+1 else l:= [l[], [h0, h1]]; h1:= h1+1; l:=[l[], [h0, h1]]; h0:=h0+1 fi od od; M:= Matrix((n+1)*n/2); for ll in l do M[ll[1], ll[2]]:= 1; M[ll[2], ll[1]]:= -1 od: isqrt(Determinant(M)); end: a:= n-> b(2*n +irem(n, 2)): seq(a(n), n=0..10); # Alois P. Heinz, May 08 2010

Mathematica

b[n_] := b[n] = Module[{l, ll, i, j, h0, h1, M}, If[n == 0 , Return[1]]; If[n<0 || MatchQ[Mod[n, 4], 1|2] , Return[0]]; l = {}; For[j = 1, j <= n-1, j++, h0 = j*(j-1)/2+1; h1 = j*(j+1)/2+1; For[i = 1, i <= j, i++, AppendTo[l, {h1, h1+1}]; If[Mod[i, 2] == 1, AppendTo[l, {h1, h0}]; h1++; AppendTo[l, {h1, h0}]; h0++ , AppendTo[l, {h0, h1}]; h1++; AppendTo[l, {h0, h1}]; h0++ ]]]; M[_, _] = 0; (M[#[[1]], #[[2]]] = 1; M[#[[2]], #[[1]]] = -1)& /@ l; Sqrt[Det[Array[M, {n*(n+1)/2, n*(n+1)/2}]]]]; a[n_] := b[2*n + Mod[n, 2]]; Table[a[n], {n, 0, 12}] (* Jean-François Alcover, Apr 29 2014, after Alois P. Heinz *)

Crossrefs

Cf. A039907, A178446, A269869.

Sequence in context: A129454 A317251 A140258 * A114045 A116899 A357193

Adjacent sequences: A071090 A071091 A071092 * A071094 A071095 A071096

Keyword

nonn

Author

N. J. A. Sloane

Extensions

a(9)-a(10) from Alois P. Heinz, May 08 2010

a(11)-a(12) from Alois P. Heinz, Jan 12 2014

Status

approved