MathDB

Consider infinite diagrams [asy] import graph; size(90); real lsf = 0.5; pen dp = linewidth(0.7) + fontsize(10); defaultpen(dp); pen ds = black; label("

n_{00} \ n_{01} \ n_{02} \ldots

", (1.14,1.38), SE*lsf); label("

n_{10} \ n_{11} \ n_{12} \ldots

", (1.2,1.8), SE*lsf); label("

n_{20} \ n_{21} \ n_{22} \ldots

", (1.2,2.2), SE*lsf); label("\vdots \vdots \qquad \vdots ", (1.32,2.72), SE*lsf); draw((1,1)--(3,1)); draw((1,1)--(1.02,2.62)); clip((-4.3,-10.94)--(-4.3,6.3)--(16.18,6.3)--(16.18,-10.94)--cycle); [/asy] where all but a finite number of the integers

n_{ij} , i = 0, 1, 2, \ldots, j = 0, 1, 2, \ldots ,

are equal to

0

. Three elements of a diagram are called adjacent if there are integers

i

and

j

with

i \geq 0

and

j \geq 0

such that the three elements are
(i)

n_{ij}, n_{i,j+1}, n_{i,j+2},

or
(ii)

n_{ij}, n_{i+1,j}, n_{i+2,j} ,

or
(iii)

n_{i+2,j}, n_{i+1,j+1}, n_{i,j+2}.

An elementary operation on a diagram is an operation by which three adjacent elements

n_{ij}

are changed into

n_{ij}'

in such a way that

|n_{ij}-n_{ij}'|=1.

Two diagrams are called equivalent if one of them can be changed into the other by a finite sequence of elementary operations. How many inequivalent diagrams exist?

Problem Statement