Sequence A002846 in <https://oeis.org/A002846> (OEIS) gives, for each positive integer $n$, the number $a(n)$ of ways of transforming a set of $n$ indistinguishable objects into $n$ singletons via a sequence of $n-1$ refinements, i.e. the number of ways of transforming a set of $n$ unlabelled objects into $n$ singletons via a sequence of $n-1$ binary partitions.


For a given $n$, each sequence of refinements can be represented by an $n\times{}n$ triangular matrix. The $(i,j)^{th}$ element of the matrix represents the number of sets after $i-1$ partitions that contain $j$ elements. The sum of the $i^{th}$ row is simply the number of sets after $i-1$ partitions, so is clearly $=i$. I am interested in linear combinations of these $a(n)$ matrices. For convenience, suppose that the coefficients total to 1. Then obviously if a matrix $M$ is formed by such a linear combination, the sum of the $i^{th}$ row is still $i$. However, a number of trials have convinced me that if $M$ is non-singular, the sum of the $i^{th}$ row of $M^{-1}$ is $\frac{i}{n}.$


To allow you to easily experiment, a list of matrices for n=4 to 9 can be obtained at github.com/helmutsimon/coalescent\_tree\_data . The file matrix\_list\_9 etc can be downloaded and then opened with pickle and gzip. The file then contains a python list of matrices represented as numpy arrays. At present, I have no idea how to go about proving this conjecture.


The following are the 4 matrices for n=5:


[[ 0. 0. 0. 0. 1.]  

 [ 1. 0. 0. 1. 0.]  

 [ 2. 0. 1. 0. 0.]  

 [ 3. 1. 0. 0. 0.]  

 [ 5. 0. 0. 0. 0.] 


[[ 0. 0. 0. 0. 1.]  

 [ 1. 0. 0. 1. 0.]  

 [ 1. 2. 0. 0. 0.]  

 [ 3. 1. 0. 0. 0.]  

 [ 5. 0. 0. 0. 0.]] 


[[ 0. 0. 0. 0. 1.]  

 [ 0. 1. 1. 0. 0.]  

 [ 2. 0. 1. 0. 0.]  

 [ 3. 1. 0. 0. 0.]  

 [ 5. 0. 0. 0. 0.]] 


[[ 0. 0. 0. 0. 1.]  

 [ 0. 1. 1. 0. 0.]  

 [ 1. 2. 0. 0. 0.]  

 [ 3. 1. 0. 0. 0.]  

 [ 5. 0. 0. 0. 0.]] 


The following are the 11 matrices for n=6: 


[[ 0. 0. 0. 0. 0. 1.]  

 [ 1. 0. 0. 0. 1. 0.]  

 [ 2. 0. 0. 1. 0. 0.]  

 [ 3. 0. 1. 0. 0. 0.]  

 [ 4. 1. 0. 0. 0. 0.]  

 [ 6. 0. 0. 0. 0. 0.]] 


[[ 0. 0. 0. 0. 0. 1.]  

 [ 1. 0. 0. 0. 1. 0.]  

 [ 2. 0. 0. 1. 0. 0.]  

 [ 2. 2. 0. 0. 0. 0.]  

 [ 4. 1. 0. 0. 0. 0.]  

 [ 6. 0. 0. 0. 0. 0.]] 


[[ 0. 0. 0. 0. 0. 1.]  

 [ 1. 0. 0. 0. 1. 0.]  

 [ 1. 1. 1. 0. 0. 0.]  

 [ 3. 0. 1. 0. 0. 0.]  

 [ 4. 1. 0. 0. 0. 0.]  

 [ 6. 0. 0. 0. 0. 0.]] 


[[ 0. 0. 0. 0. 0. 1.]  

 [ 1. 0. 0. 0. 1. 0.]  

 [ 1. 1. 1. 0. 0. 0.]  

 [ 2. 2. 0. 0. 0. 0.]  

 [ 4. 1. 0. 0. 0. 0.]  

 [ 6. 0. 0. 0. 0. 0.]] 


[[ 0. 0. 0. 0. 0. 1.]  

 [ 0. 1. 0. 1. 0. 0.]  

 [ 2. 0. 0. 1. 0. 0.]  

 [ 3. 0. 1. 0. 0. 0.]  

 [ 4. 1. 0. 0. 0. 0.]  

 [ 6. 0. 0. 0. 0. 0.]] 


[[ 0. 0. 0. 0. 0. 1.]  

 [ 0. 1. 0. 1. 0. 0.]  

 [ 2. 0. 0. 1. 0. 0.]  

 [ 2. 2. 0. 0. 0. 0.]  

 [ 4. 1. 0. 0. 0. 0.]  

 [ 6. 0. 0. 0. 0. 0.]] 


[[ 0. 0. 0. 0. 0. 1.]  

 [ 0. 1. 0. 1. 0. 0.]  

 [ 1. 1. 1. 0. 0. 0.]  

 [ 3. 0. 1. 0. 0. 0.]  

 [ 4. 1. 0. 0. 0. 0.]  

 [ 6. 0. 0. 0. 0. 0.]] 


[[ 0. 0. 0. 0. 0. 1.]  

 [ 0. 1. 0. 1. 0. 0.]  

 [ 1. 1. 1. 0. 0. 0.]  

 [ 2. 2. 0. 0. 0. 0.]  

 [ 4. 1. 0. 0. 0. 0.]  

 [ 6. 0. 0. 0. 0. 0.]] 


[[ 0. 0. 0. 0. 0. 1.]  

 [ 0. 1. 0. 1. 0. 0.]  

 [ 0. 3. 0. 0. 0. 0.]  

 [ 2. 2. 0. 0. 0. 0.]  

 [ 4. 1. 0. 0. 0. 0.]  

 [ 6. 0. 0. 0. 0. 0.]] 


[[ 0. 0. 0. 0. 0. 1.]  

 [ 0. 0. 2. 0. 0. 0.]  

 [ 1. 1. 1. 0. 0. 0.]  

 [ 2. 2. 0. 0. 0. 0.]  

 [ 4. 1. 0. 0. 0. 0.]  

 [ 6. 0. 0. 0. 0. 0.]] 


[[ 0. 0. 0. 0. 0. 1.]  

 [ 0. 0. 2. 0. 0. 0.]  

 [ 1. 1. 1. 0. 0. 0.]  

 [ 3. 0. 1. 0. 0. 0.]  

 [ 4. 1. 0. 0. 0. 0.]  

 [ 6. 0. 0. 0. 0. 0.]]



OEIS Sequence A002846 and properties of matrix inverses
Actually, this turned out to be embarrassingly simple. Let $M$ be such an $n\times{}n$ matrix and set $\mathbf{i}\_n$ to be the column matrix with entries 1 to $n$ and $\mathbf{1}\_n$ to be the column matrix with entries all equal to 1. By construction, each row of the matrix represents a partition of $n$. The first element of the row is the number of partitions of block size 1, the second element is the number of partitions of block size 2 and so on. So
$$M.\mathbf{i}\_n=n\mathbf{1}\_n$$
So if $M^{-1}$ exists:
$$M^{-1}.M.\mathbf{i}\_n=M^{-1}.n\mathbf{1}\_n$$
$$M^{-1}\mathbf{1}\_n=\frac{\mathbf{i}\_n}{n}$$


