# Revision history [back]

The oeis is here: A006982

It is a highly combinatorial job to compute the numbers in the sequence.

Their strategy is to compute first the numbers of the vertically indecomposable, distributive lattices. Which are lattices that cannot be decomposed as a vertical sum of at least two distributive lattices. The vertical sum of $$(D_A;0_A,1_A;\le_A)$$ (placed in our minds vertically below) and $$(D_B;0_A,1_A;\le_A)$$ (placed in our minds vertically above the previous structure) is the lattice on the set $D=D_A\sqcup D_B/1_A\sim 0_B$ with $0=0_A$ and $1=1_B$ and the induced order that makes $D_A\le D_B$, elementwise.

Each distributive lattice appears uniquely as a vertical sum of vertically independent distributive lattices. The generating series for the vertically independent ones is: $V(t) = t^2+t^4+t^6+3t^8+t^9+6t^{10}+2t^{11}+16t^{12}+8t^{13}+42t^{10}+28t^{15}+112t^{16}+\dots$ and form here on the numbers start to grow "also for the human eye in exponential taste". The coefficient in power $t^{49}$ is $V_{49}= 5 045 200 597$. Having this list, we can generate the list for the distribute lattices, we will get $$D = \sum_{k\ge 0}D_kt^k$$ with $D_{49}=908 414 736 485$.

At the level of enumeration, if we know the coefficients in $V$, then we can find the ones in $D$ simply. Let $W=V/t$. Then $$D=t(1+V+V^2+\dots)=\frac t{1-V}$$ with computations in $\mathbb Z[[ t ]]$ . Sage code for this:

R.<t> = PolynomialRing( QQ )
V = ( t^2 + t^4 + t^6 + 3*t^8 + t^9
+ 6*t^10 + 2*t^11 + 16*t^12 + 8*t^13 + 42*t^14 + 28*t^15 + 112*t^16 + 93*t^17 + 311*t^18 + 295*t^19
+ 869*t^20 + 939*t^21 + 2454*t^22 + 2931*t^23 + 7032*t^24 + 9036*t^25 + 20301*t^26 + 27701*t^27 + 58929*t^28 + 84413*t^29
+ O(t^30) )
W = V/t
D = t/(1-W)
for c in D.coefficients():
print c


This gives:

1
1
1
2
3
5
8
15
26
47
82
151
269
494
891
1639
2978
5483
10006
18428
33749
62162
114083
210189
386292
711811
1309475
2413144
4442221


In accordance with A006982.

So it is enough to write code that generates the vertically independent, distributive lattices.

I will try to do this till degree $64$ in the following days... This is then computing $D$ also till degree $64$.