A358069 Number of configurations of the 20 Vertex model on a square grid n X n with domain wall boundary conditions.

1, 3, 23, 433, 19705, 2151843, 561696335, 349667866305, 518369549769169, 1828200035691135203, 15328648070256551849383, 305390661137273761896820529, 14451387790147329024372260663689, 1623803344366103974773282069705064899, 433134712202745984875469054553527204825375

Offset

1,2

Comments

Also number of quarter-turn symmetric domino tilings of a holey Aztec quasi-square.

Links

Table of n, a(n) for n=1..15.

Paul Barry, A Riordan array family for some integrable lattice models, arXiv:2409.09547 [math.CO], 2024. See p. 8.

P. Di Francesco and E. Guitter, Twenty-Vertex model with domain wall boundaries and domino tilings, Electr. J. Combin. 27-2 (2020), P2.13.

Formula

a(n) = det A(n), where A(n) is n X n matrix (a[i,j]), 1 <= i,j <= n, with a[i,j] = [x^(i-1)*y^(j-1)] 1/(1-x*y)+2*x/((1-x)*(1-x-y-x*y)).

Maple

g:=  1/(1-x*y)+2*x/((1-x)*(1-x-y-x*y)):
S:= map(series, series(g, x, 21), y, 21):
A:= Matrix(20, 20, (i, j) -> coeff(coeff(S, x, i), y, j)):
1, seq(LinearAlgebra:-Determinant(A[1..n, 1..n]), n=1..20); # Robert Israel, Dec 18 2022

Mathematica

a[n_]:=Det[Table[KroneckerDelta[i, j] + Coefficient[Normal[Series[(2 x)/((1 - x) (1 - x - y - x y)) x y, {x, 0, n}, {y, 0, n}]], x^(i) y^(j)], {i, 1, n}, {j, 1, n}]]; Array[a, 15]

Crossrefs

Sequence in context: A271851 A133338 A298466 * A116986 A271962 A268668

Adjacent sequences: A358066 A358067 A358068 * A358070 A358071 A358072

Keyword

nonn

Author

Philippe Di Francesco, Dec 17 2022

Status

approved