Counting Total Number of Non-Equivalent Configurations in a 2-D Grid

Question

This is a challenging question I've been trying (unsuccessfully) to solve via programming, math or both.

Suppose you're given a 2D grid, whose width and height, $w$ and $h$, can each range from $1$ thru $12$ inclusive. Each cell in this grid can be in any of $k$ states (where $k$ ranges from $2$ thru $20$).

The problem is to find the total number of distinct (non-equivalent) state configurations of this grid.

Definition: Two configurations of a grid, $c_1$ and $c_2$, are deemed "equivalent" if it is possible to change $c_1$ to $c_2$ by swapping any pair of rows and/or any pair of columns (you can perform the swap operation as many times as you want).

To take an example, let's consider a grid with $w=2$, $h=2$, and $k=2$. We will represent each of the two states as $0$ and $1$ respectively.

One possible configuration of the grid is one where all the cells are in state $0$:

00
00

You could swap any rows and/or columns without changing anything. So the above counts as 1 configuration.

Another configuration is where only 1 cell is in state $1$. There are 4 possible arrangements:

01 10 00 00
00 00 01 10

But note that all of the 4 arrangements are "equivalent", because you can get any of the arrangements by swapping rows and/or columns. So the above counts as 1 configuration.

Next, exactly 2 cells are in state $1$:

11 00
00 11

Another (non-equivalent) way in which 2 cells are in state $1$:

10 01
10 01

And yet another:

10 01
01 10

So there are 3 non-equivalent configurations in which exactly 2 cells are in state $1$.

Next, exactly 3 cells are in state $1$:

10 01 11 11
11 11 10 01

1 configuration, just as in the case where only 1 cell was in state $1$.

Finally, all 4 cells are in state $1$:

11
11

1 configuration, just as in the case where all 4 cells were in state $0$.

So, counting them up, there are exactly $7$ distinct configurations for the case where $w=2, h=2, k=2$.

For reference, here are some brute-force test cases I ran on some additional examples (where the w,h,k values are fairly small):

w=2,h=1,k=1 Answer: 1
w=2,h=1,k=2 Answer: 3
w=2,h=1,k=3 Answer: 6
w=2,h=1,k=4 Answer: 10
w=2,h=1,k=5 Answer: 15
w=2,h=1,k=6 Answer: 21
...
So at least when w, h are fixed at 1,2 (or at 2,1), the pattern seems pretty obvious as k increases. Here is another suite of test cases for w=2, h=2:

w=2,h=2,k=1 Answer: 1
w=2,h=2,k=2 Answer: 7
w=2,h=2,k=3 Answer: 27
w=2,h=2,k=4 Answer: 76
w=2,h=2,k=5 Answer: 175
w=2,h=2,k=6 Answer: 351
...
This is much less obvious to me. If there is a pattern, I'm unsure how to find it.

One more suite of test cases for good measure:

w=2,h=3,k=1 Answer: 1
w=2,h=3,k=2 Answer: 13
w=2,h=3,k=3 Answer: 92
w=2,h=3,k=4 Answer: 430
w=2,h=3,k=5 Answer: 1505
...
yep I'm lost as to what insight if any can be gained.

So, for some arbitrary $w,h,k$, I'm wondering whether there is some type of closed-form formula or perhaps a way to get the answer via dynamic programming/recursion or perhaps divide-and-conquer? I've tried a few of these approaches and didn't really see any obvious breakthrough. The answers get very large very quickly (for example, for $w=2,h=3,k=4$, the answer is 430). I can only imagine how large the answer would be for, say, $w=12,h=12,k=20$.

I would appreciate any help on this. Thanks!

Answer 1

We will use Burnside's lemma, plus some additional optimizations.

Note that row swaps and column swaps commute, so the group of allowable transformations to the grid is exactly $G = S_w \times S_h$, where $S_n$ is the symmetric group on $n$ symbols. Note that $G$ embeds in $S_{wh}$ via a canonical embedding $\varphi : G \to S_{wh}$, given by $\varphi(\pi,\sigma) = \tau$ where $\tau$ is defined by $\tau(i,j) = (\pi(i), \sigma(j))$. Therefore, we will think of $G$ simultaneously as the direct product $S_w \times S_h$ and also as a subgroup of $S_{wh}$.

Now let's count the number of grids that are fixed by a group element $g=(\pi,\sigma) \in G$. Define the cycle type of a permutation $\pi$ to be the multiset of lengths of the cycles in the cycle decomposition of $\pi$. If $\pi \in S_n$, the cycle type of $\pi$ is a partition of $n$. With this in mind, we will let $\ell(\pi)$ denote the cycle type of $\pi \in S_w$ and $\ell(\sigma)$ the cycle type of $\sigma \in S_h$. Finally, let $\ell((\pi,\sigma))$ denote the cycle type of $(\pi,\sigma)$, thinking of $(\pi,\sigma)$ as an element of $S_{wh}$.

We can express $\ell((\pi,\sigma))$ in terms of $\ell(\pi)$ and $\ell(\sigma)$. Let $x,y$ range over all $x \in \ell(\pi), y \in \ell(\sigma)$. For each $x,y$, $\ell((\pi,\sigma))$ has $\gcd(x,y)$ copies of $\operatorname{lcm}(x,y)$. It follows that

$$|\ell((\pi,\sigma))| = \sum_{x \in \ell(\pi)} \sum_{y \in \ell(\sigma)} m_x n_y \gcd(x,y),$$

where $x,y$ range over all distinct values (without repetition), and $m_x$ is the multiplicity of $x$ in $\ell(\pi)$ (the number of times $x$ appears in $\ell(\pi)$) and $n_y$ is the multiplicity of $y$ in $\ell(\sigma)$.

Now the number of grids that are fixed by $(\pi,\sigma)$ is

$$f(\pi,\sigma) = k^{|\ell((\pi,\sigma))|},$$

where $|\ell((\pi,\sigma))|$ counts the number of cycles in $(\pi,\sigma) \in G$ (multiplicities are counted). By the above formula for $|\ell((\pi,\sigma))|$, we see that

$$f(\pi,\sigma) = k^{\sum_{x \in \ell(\pi)} \sum_{y \in \ell(\sigma)} m_x n_y \gcd(x,y)}.$$

To apply Burnside's lemma, it remains to sum $f(g)$ over all elements $g \in G$, as Burnside's lemma tells us that the total number of non-equivalent grids is

$$N = {1 \over |G|} \sum_{g \in G} f(g) = {1 \over w! h!} \sum_{g \in G} f(g).$$

In principle, we could compute this sum by enumerating all $w! h!$ elements in $G$, then evaluating $f$ on each, and summing the result.

It turns out this can be optimized further, as $f(\pi,\sigma)$ depends only on $\ell(\pi)$ and $\ell(\sigma)$, but not on any other aspect of $\pi$ or $\sigma$. Thus, we can more efficiently evaluate the sum using the formula

$$\sum_{g \in G} f(g) = \sum_{P,Q} N(P,Q) \times k^{\sum_{x \in P} \sum_{y \in Q} \gcd(x,y)},$$

where the inner sum ranges over all partitions $P$ of $w$ and all partitions $Q$ of $h$, and where we define

$$N(P,Q) = |\{(\pi,\sigma) \in G : \ell(\pi)=P, \ell(\sigma)=Q\}|$$

to count the number of group elements of $G$ whose cycle types are $P,Q$. Now we have

$$N(P,Q) = N_w(P) \times N_h(Q),$$

where $N_w(P) = |\{\pi \in S_w : \ell(\pi)=P\}|$ is the number of permutations in $S_w$ with cycle type $P$, and similarly $N_h(Q)$ is the number of permutations in $S_h$ with cycle type $Q$. Finally, there is a known formula to compute $n_w(P)$ for any desired partition $P$; it is

$$N_w(P) = {w! \over \prod_{x \in P} m_x! x^{m_x}},$$

where $x$ ranges over the distinct elements of $P$ and $m_x$ is the multiplicity of $x$ in $P$ (the number of times $x$ appears in $P$).

Plugging in, we find that the total number of non-equivalent grids is given by

$$N = \sum_{P,Q} {k^{\sum_{x \in P} \sum_{y \in Q} m_x n_y \gcd(x,y)} \over \prod_{x \in P} m_x! x^{m_x} \prod_{y \in Q} n_y! y^{n_y}}.$$

As before, in this formula $x,y$ range over distinct values, $m_x$ is the multiplicity of $x$ in $P$, and $n_y$ is the multiplicity of $y$ in $Q$.

This can be computed by two nested for-loops that iterate over all partitions $P$ of $w$ and all partitions $Q$ of $h$.

What is the running time of this algorithm? The asymptotic running time is approximately $O(p(w) p(h))$, where $p(n)$ is the number of partitions of the integer $n$. It is known that asymptotically $p(n) = O(\exp\{\pi \sqrt{2n/3} \})$, so it follows that the asymptotic running time of this algorithm is approximately $O(\exp\{\pi \sqrt{2/3} (\sqrt{w}+\sqrt{h})\})$, or equivalently, approximately $O(2^{3.7(\sqrt{w}+\sqrt{h})})$. This should allow you to handle the full range of values $1 \le w,h \le 12$, as for that range we have $\sqrt{w}+\sqrt{h} < 7$ and $2^{3.7 \times 7} \approx 2^{26}$ is not too large.