## Zimin Words and Bifixes

One of the earliest contributions to the On-Line Encyclopedia of Integer Sequences (OEIS) was a family sequences counting the number of words that begin (or don’t begin) with a palindrome:

• Let $$f_k(n)$$ be the number of strings of length $$n$$ over a $$k$$-letter alphabet that begin with a nontrivial palindrome” for various values of $$k$$.
• Let $$g_k(n)$$ be the number of strings of length n over a $$k$$-letter alphabet that do not begin with a nontrivial palindrome.
• Number of binary strings of length $$n$$ that begin with an odd-length palindrome. (A254128)

(If I had known better, I would have published fewer sequences in favor of a table, and I would have requested contiguous blocks of A-numbers.)

I must have written some Python code to compute some small terms of this sequence, and I knew that $$g_k(n) = k^n – f_k(n)$$, but I remember being at in my friend Q’s bedroom when the recursion hit me for $$f_k(n)$$: $$f_k(n) = kf_k(n-1) + k^{\lceil n/2 \rceil} – f_k\big(\lceil \frac n 2 \rceil \big)$$

## “Bifix-free” words

One sequence that I didn’t add to the OEIS was the “Number of binary strings of length n that begin with an even-length palindrome”—that’s because this was already in the Encyclopedia under a different name:

A094536: Number of binary words of length n that are not “bifix-free”.

0, 0, 2, 4, 10, 20, 44, 88, 182, 364, 740, 1480, 2980, 5960, …

A “bifix” is a shared prefix and suffix, so a “bifix-free” word is one such that all prefixes are different from all suffixes. More concretely, if the word is $$\alpha_1\alpha_2 \dots \alpha_n$$, then $$(\alpha_1, \alpha_2, \dots, \alpha_k) \neq (\alpha_{n-k+1},\alpha_{n-k+2},\dots,\alpha_n)$$ for all $$k \geq 1$$.

The reason why the number of binary words of length $$n$$ that begin with an even length palindrome is equal to the number of binary words of length $$n$$ that have a bifix is because we have a bijection between the two sets. In particular, find the shortest palindromic prefix, cut it in half, and stick the first half at the end of the word, backward. I’ve asked for a better bijection on Math Stack Exchange, so if you have any ideas, please share them with me!

In 2019–2020, Daniel Gabric, Jeffrey Shallit wrote a paper closely related to this called Borders, Palindrome Prefixes, and Square Prefixes.

## Zimin words

A Zimin word can be defined recursively, but I think it’s most suggestive to see some examples:

• $$Z_1 = A$$
• $$Z_2 = ABA$$
• $$Z_3 = ABACABA$$
• $$Z_4 = ABACABADABACABA$$
• $$Z_n = Z_{n-1} X Z_{n-1}$$

All Zimin words $$Z_n$$ are examples of “unavoidable patterns”, because every sufficiently long string with letters in any finite alphabet contains a substring that matches the $$Z_n$$ pattern.

For example the word $$0100010010111000100111000111001$$ contains a substring that matches the Zimin word $$Z_3$$. Namely, let $$A = 100$$, $$B = 0$$, and $$C = 1011$$, visualized here with each $$A$$ emboldened: $$0(\mathbf{100}\,0\,\mathbf{100}\,1011\,\mathbf{100}\,0\,\mathbf{100})111000111001$$.

I’ve written a Ruby script that generates a random string of length 29 and uses a regular expression to find the first instance of a substring matching the pattern $$Z_3 = ABACABA$$. You can run it on TIO, the impressive (and free!) tool from Dennis Mitchell.

# Randomly generates a binary string of length 29.
random_string = 29.times.map { [0,1].sample }.join("")
p random_string
# Finds the first Zimin word ABACABA
p random_string.scan(/(.+)(.+)\1(.+)\1\2\1/)[0]
# Pattern:             A   B   A C   A B A


Why 29? Because all binary words of length 29 contain the pattern $$Z_3 = ABACABA$$. However, Joshua Cooper and Danny Rorabaugh’s paper provides 48 words of length 28 that avoid that pattern (these and their reversals):

1100000010010011011011111100
1100000010010011111101101100
1100000010101100110011111100
1100000010101111110011001100
1100000011001100101011111100
1100000011001100111111010100
1100000011011010010011111100
1100000011011011111100100100
1100000011111100100101101100
1100000011111100110011010100
1100000011111101010011001100
1100000011111101101100100100

1100100100000011011011111100
1100100100000011111101101100
1100100101101100000011111100
1100110011000000101011111100
1100110011000000111111010100
1100110011010100000011111100
1101010000001100110011111100
1101010000001111110011001100
1101010011001100000011111100
1101101100000010010011111100
1101101100000011111100100100
1101101100100100000011111100

## The Zimin Word $$Z_2 = ABA$$ and Bifixes

The number of Zimin words of length $$n$$ that match the pattern ABA is equal to the number of of words that begin with an odd-length palindrome. Analogously, the number of words with a bifix is equal to the number of words that begin with an even-length palindrome. The number of these agree when $$n$$ is odd.

I’ve added OEIS sequences A342510A342512 which relate to how numbers viewed as binary strings avoid—or fail to avoid—Zimin words. I asked users to implement this on Code Golf Stack Exchange.

## Richard Guy’s Partition Sequence

Neil Sloane is the founder of the On-Line Encyclopedia of Integer Sequences (OEIS). Every year or so, he gives a talk at Rutgers in which he discusses some of his favorite recent sequences. In 2017, he spent some time talking about a 1971 letter that he got from Richard Guy, and some questions that went along with it. In response to the talk, I investigated the letter and was able to sort out Richard’s 45-year-old idea, and correct and compute some more terms of his sequence.

## Richard Guy and his sequences

Richard Guy was a remarkable mathematician who lived to the remarkable age of 103 years, 5 months, and 9 days! His life was filled with friendships and collaborations with many of the giants of recreational math: folks like John Conway, Paul Erdős, Martin Gardner, Donald Knuth, and Neil Sloane. But what I love most about Richard is how much joy and wonder he found in math. (Well, that and his life-long infatuation with his wife Louise.)

[I’m] an amateur [mathematician], I mean I’m not a professional mathematician. I’m an amateur in the more genuine sense of the word in that I love mathematics and I would like everybody in the world to like mathematics.

Richard Guy in Fascinating Mathematical People: Interviews and Memoirs

### Richard’s letter to Neil

In January 2017, Neil Sloane gave a talk at Doron Zeilberger’s Experimental Mathematics Seminar, and about six minutes in, Neil discusses a letter that Richard sent to him at Cornell—which was the forwarded to Bell Labs—in June 1971.

When I was working on the book, the 1973 Handbook of Integer Sequences, I would get letters from Richard Guy from all over the world. As he traveled around, he would collect sequences and send them to me.

Neil Sloane, Rutgers Experimental Mathematics Seminar

At 11:30, Neil discusses “sequence I” from Richard’s letter, which he added to the OEIS as sequence A279197:

Number of self-conjugate inseparable solutions of $$X + Y = 2Z$$ (integer, disjoint triples from $$\{1,2,3,\dots,3n\}$$).

Neil mentioned in the seminar that he didn’t really know exactly what the definition meant. With some sleuthing and programming, I was able to make sense of the definition, write a Haskell program, correct the 7th term, and extend the sequence by a bit. The solutions for $$A279197(1)$$ through $$A279197(10)$$ are listed in a file I uploaded to the OEIS, and Fausto A. C. Cariboni was able to extend the sequence even further, submitting terms $$A279197(11)$$–$$A279197(17)$$.

## How the sequence works.

The idea here is to partition $$\{1,2,3,\dots,3n\}$$ into length-3 arithmetic progressions, $$\bigl\{\{X_i,Z_i,Y_i\}\bigr\}_{i=1}^{n}$$. And in particular, we want them to be inseparable and self-conjugate.

An inseparable partition is one whose “smallest” subsets are not a solution for a smaller case. For example, if $$n=3$$, then the partition $\bigl\{ \{1,3,5\}, \{2,4,6\}, \{7,8,9\} \bigr\}$ is separable, because if the subset $$\bigl\{ \{1,3,5\}, \{2,4,6\} \bigr\}$$ is a solution to the $$n=2$$ case.

A self-conjugate partition is one in which swapping each $$i$$ with each $$3n+1-i$$ gets back to what we started with. For example, $$\bigl\{\{1,3,5\}, \{2,4,6\}\bigr\}$$ is self-congugate, because if we replace the $$1$$ with a $$6$$ and the $$2$$ with a $$5$$, and the $$i$$ with a $$7-i$$, then we get the same set: $$\bigl\{\{6,4,2\}, \{5,3,1\} \bigr\}$$

### Generalizing Richard Guy’s idea

Of course, it’s natural to wonder about the separable solutions, or what happens if the self-conjugate restriction is dropped. In exploring these cases, I found four cases already in the OEIS, and I computed five more: A282615A282619.

## Generalizing further

There are lots of other generalizations that might be interesting to explore. Here’s a quick list:

• Look at partitions of $$\{1,2,\dots,kn\}$$ into $$n$$ parts, all of which are an arithmetic sequence of length $$k$$.
• Count partitions of $$\{1,2,\dots,n\}$$ into any number of parts of (un)equal size in a way that is (non-)self-conjugate and/or (in)separable.
• Consider at partitions of $$\{1,2,\dots,3n\}$$ into $$n$$ parts, all of which are an arithmetic sequence of length $$3$$, and whose diagram is “non-crossing”, that is, none of the line segments overlap anywhere. (See the 6th and 11th cases in the example for $$A279197(6) = 11$$.)

If explore any generalizations of this problem on your own, if you’d like to explore together, or if you have an anecdotes about Richard Guy that you’d like to share, let me know on Twitter!

## Polytopes with Lattice Coordinates

Problems 21, 66, and 116 in my Open Problem Collection concern polytopes with lattice coordinates—that is, polygons, polyhedra, or higher-dimensional analogs with vertices the square or triangular grids. (In higher dimensions, I’m most interested in the $$n$$-dimensional integer lattice and the $$n$$-simplex honeycomb).

This was largely inspired by one of my favorite mathematical facts: given a triangular grid with $$n$$ points per side, you can find exactly $$\binom{n+2}{4}$$ equilateral triangles with vertices on the grid. However, it turns out that there isn’t a similarly nice polynomial description of tetrahedra in a tetrahedron or of triangles in a tetrahedron. (Thanks to Anders Kaseorg for his Rust program that computed the number of triangles in all tetrahedra with 1000 or fewer points per side.)

The $$4$$-simplex (the $$4$$-dimensional analog of a triangle or tetrahedron) with $$n-1$$ points per side, has a total of $$\binom{n+2}{4}$$ points, so there is some correspondence between points in some $$4$$-dimensional polytope, and triangles in the triangular grid. This extends to other analogs of this problem: the number of squares in the square grid is the same as the number of points in a $$4$$-dimensional pyramid.

## The $$\binom{n+2}{4}$$ equilateral triangles

I put a Javascript applet on my webpage that illustrates a bijection between size-$$4$$ subsets of $$n+2$$ objects and triangles in the $$n$$-points-per-side grid. You can choose different subsets and see the resulting triangles. (The applet does not work on mobile.)

## Polygons with vertices in $$\mathbb{Z}^n$$

This was also inspired by Mathologer video “What does this prove? Some of the most gorgeous visual ‘shrink’ proofs ever invented”, where Burkard Polster visually illustrates that the only regular polygons with vertices in $$\mathbb{Z}^n$$ (and thus the $$n$$-simplex honeycomb) are equilateral triangles, squares, and regular hexagons.

## Polyhedra with vertices in $$\mathbb{Z}^3$$

There are some surprising examples of polyhedra in the grid, including cubes with no faces parallel to the $$xy$$-, $$xz$$-, or $$yz$$-planes.

While there are lots of polytopes that can be written with vertices in $$\mathbb{Z}^3$$, Alaska resident and friend RavenclawPrefect cleverly uses Legendre’s three-square theorem to prove that there’s no way to write the uniform triangular prism this way! However, he provides a cute embedding in $$\mathbb{Z}^5$$: the convex hull of $$\scriptsize{\{(0,0,1,0,0),(0,1,0,0,0),(1,0,0,0,0),(0,0,1,1,1),(0,1,0,1,1),(1,0,0,1,1)}\}.$$

## Polygons on a “centered $$n$$-gon”

I asked a question on Math Stack Exchange, “When is it possible to find a regular $$k$$-gon in a centered $$n$$-gon“—where “centered $$n$$-gon” refers to the diagram that you get when illustrating central polygonal numbers. These diagrams are one of many possible generalizations of the triangular, square, and centered hexagonal grids. (Although it’s worth noting that the centered triangular grid is different from the ordinary triangular grid.)

## A catalog of polytopes and grids

On my OEIS wiki page, I’ve created some tables that show different kinds of polytopes in different kinds of grids. There are quite a number of combinations of polygons/polyhedra and grids that either don’t have an OEIS sequence or that I have been unable to find.

If you’re interested in working on filling in some of the gaps in this table, I’d love it if you let me now! And if you’d like to collaborate or could use help getting started, send me a message on Twitter!

## Parity Bitmaps from the OEIS

My friend Alec Jones and I wrote a Python script that takes a two-dimensional sequence in the On-Line Encyclopedia of Integer Sequences and uses it to create a one-bit-per-pixel (1BPP) “parity bitmaps“. The program is simple: it colors a given pixel is black or white depending on whether the corresponding value is even or odd.

### An Unexpected Fractal

We’ve now run the script on over a thousand sequences, but we still both agree on our favorite: the fractal generated by OEIS sequence A279212.

Fill an array by antidiagonals upwards; in the top left cell enter $$a(0)=1$$; thereafter, in the $$n$$-th cell, enter the sum of the entries of those earlier cells that can be “seen” from that cell.

Notice that in the images below, increasing the rows and columns by a factor of $$2^n$$ seems to increase the “resolution”, because the parity bitmap is self similar at 2x the scale. We still don’t have a good explanation for why we’d expect these images are fractals. If you know, please answer our question about it on Math Stack Exchange. (Alec and I have generated these images up to 16384 × 32768 resolution, roughly 536 megapixels.)

#### The Construction of the Sequence

The sequence is built up by “antidiagonals”, as shown in the GIF below. In the definition, “seen” means every direction a chess queen can move that already has numbers written down (i.e. north, west, northwest, or southwest). That is, look at all of the positions you can move to, add them all up, write that number in your square, move to the next square, and repeat. (The number in cell $$C$$ also counts the number of paths a queen can make from $$C$$ to the northwest corner using only N, NW, W, SW moves.)

(Interestingly, but only tangentially related: Code Golf Stack Exchange User flawr noticed that the number of north/west rook walks is related to the number of ways of partitioning a $$1 \times n$$ grid into triangles.)

## Parity Bitmaps for Other Sequences

It’s worth noting that many sequences are all black, consist of simple repeating patterns, or look like static. However, chess-type constructions, as illustrated by the GIF above, the one above yield images that look like the Sierpiński triangle. (See A132439 and A334017 below, and look at A334016 and A334745 via their OEIS entries.) Look below for a couple other sequences with interesting images too.

I ordered a poster-sized print of the A279212 fractal for Alec, and he framed it in his office.

Some ideas for further exploration:

## Stacking LEGO Bricks

Back in May, I participated in The Big Lock-Down Math-Off from The Aperiodical. In the Math-Off, I went head-to-head against Colin Beveridge (who has, hands-down, my favorite Twitter handle: @icecolbeveridge). Colin wrote about using generating functions to do combinatorics about Peter Rowlett’s toy Robot Caterpillar. Coincidentally and delightfully, I wrote about using generating functions to do combinatorics about Peter Kagey’s toy LEGOs.

Counting LEGO configurations is a problem dating back to at least 1974, when Jørgen Kirk Kristiansen counted that there are 102,981,500 ways to stack six 2×4 LEGOs of the same color into a tower of height six. According to Søren Eilers, Jørgen undercounted by 4!

In my Math-Off piece, I wrote about a fact that I learned from Math Stack Exchange user N. Shales—a fact that may be my (and Doron Zeilberger’s) favorite in all of mathematics: there are exactly $$3^{n-1}$$ ways to make a tower out of $$1 \times 2$$ LEGO bricks following some simple and natural rules. Despite this simple formula, the simplest known proof is relatively complicated and uses some graduate-level combinatorial machinery.

## The Rules

1. The bricks must lie in a single plane.
2. No brick can be directly on top of any other.
3. The bottom layer must be a continuous strip.
4. Every brick that’s not on the bottom layer must have at least one brick below it.

Gouyou-Beauchamps and Viennot first proved this result in their 1988 statistical mechanics paper, but the nicest proof that I know of can be found in Miklós Bóna’s Handbook of Enumerative Combinatorics (page 26). Bóna’s proof decomposes the stacks of blocks in a clever way and then uses a bit of generating function magic.

## Other rules

In preparation for the Math-Off piece, I asked a question on Math Stack Exchange about counting the number of towers without Rule 2. The user joriki provided a small and delightful modification of Bóna’s proof that proves that there are $$4^{n-1}$$ towers if only rules 1, 3, and 4 are followed.

It might also be interesting to consider the 14 other subsets of the rules. I encourage you to compute the number of corresponding towers and add any new sequences to the On-Line Encyclopedia of Integer Sequences. If you do so, please let me know! And I’d be happy to work with you if you’d like to contribute to the OEIS but don’t know how to get started.

Another natural question to ask: How many different towers can you build out of $$n$$ bricks if you consider mirror images to be the same? In the example above with the red bricks, there are six different towers, because there are three pairs of mirror images. By Burnside’s Lemma (or a simpler combinatorial argument) this is equivalent to counting the number of symmetric brick stacks. If there are $$s(n)$$ symmetric towers with $$n$$ bricks, then there are $$\displaystyle \frac 12 (3^{n-1}+s(n))$$ towers. For $$n = 4$$, there are three such towers, shown below in blue.

I asked about this function on Math Stack Exchange and wrote a naive Haskell program to compute the number of symmetric towers consisting of $$n \leq 19$$ bricks, which I added to the OEIS as sequence A320314. OEIS contributor Andrew Howroyd impressively extended the sequence by 21 more terms. I also added sequence $$A264746 = \frac 12 (3^{n-1}+A320314(n))$$, which counts towers up to reflection, and A333650, which is a table that gives the number of towers with $$n$$ bricks and height $$k$$.

## Stacking Ordinary Bricks

It is also interesting to count the number of (stable) towers that can be made out of ordinary bricks without any sort of mortar. I asked on Math Stack Exchange for a combinatorial rule for determining when a stack of ordinary bricks is stable. MSE user Jens commented that this problem is hard, and pointed to the OEIS sequence A168368 and the paper “Maximum Overhang” by Mike Paterson, Yuval Peres, Mikkel Thorup, Peter Winkler, and Uri Zwick, which provides a surprising example of a tower that one might expect to be stable, but in fact is not.

I’d still like to find a combinatorial rule, or implement a small physics engine, to determine when a stack of bricks is stable.

These problems and some generalizations can be found in Problem 33 of my Open Problem Collection. If you’d like to collaborate on any of these problems, let me know on Twitter. If you find yourself working on your own, I’d love for you to keep me updated with your progress!

(The graphics of LEGO bricks were rendered using the impressive and free LEGO Studio from BrickLink.)

## Regular Truchet Tilings

I recently made my first piece of math art for my apartment: a 30″×40″ canvas print based on putting Truchet tiles on the truncated trihexagonal tiling.

I first became interested in these sorts of patterns after my former colleague Shane sent me a YouTube video of the one-line Commodore 64 BASIC program:
 10 PRINT CHR$(205.5+RND(1)); : GOTO 10  I implemented a version of this program on my website, with the added feature that you could click on a section to recolor the entire component, and this idea was also the basis of Problem 2 and Problem 31 in my Open Problem Collection. I saw this idea generalized by Colin Beveridge in the article “Too good to be Truchet” in Chalkdust Magazine. In this article, Colin counts the ways of drawing hexagons, octagons, and decagons with curves connecting the midpoints of edges, and regions colored in an alternating fashion. In the case of the hexagon, there are three ways to do it, one of which looks like Palago tiles. It turns out that if you ignore the colors, the number of ways to pair up midpoints of the sides of a$latex 2n$-gon in such a way that the curves connecting the midpoints don’t overlap is given by the$latex n$-th Catalan number. For example, there are$latex C_4 = 14$ways of connecting midpoints of the sides of an octagon, where different rotations are considered distinct. There are three regular tilings of the plane by$latex 2n$-gons, the square tiling, the truncated square tiling, and the truncated trihexagonal tiling. Placing a Truchet tile uniformly at random over each of the$latex 2n\$-gons, results in a really lovely emergent structure.

If you find these designs as lovely as I do, I’d recommend taking a look at the Twitter bots @RandomTiling by Dave Richeson and @Truchet_Nested/@Trichet_Nested by @SerinDelaunay (based on a idea from Christopher Carlson) which feature a series of visually interesting generalizations of Truchet tilings and which are explained in Christopher’s blog post “Multi-scale Truchet Patterns“.

Edward Borlenghi has a blog post “The Curse of Truchet’s Tiles” about how he tried—mostly unsuccessfully—to sell products based on Truchet tiles, like carpet squares and refrigerator magnets (perhaps similar to “YoYo” magnets from Magnetic Poetry). The post is filled with lots of cool, alternative designs for square Truchet tiles and how they fit together. Edward got a patent for some of his ideas, and his attempt to sell these very cool products feels like it could have been my experience in another life.

If you want to see more pretty images and learn more about this, make sure to read Truchet Tilings Revisited by Robert J. Krawczyk! If you want to see what this looks like on a spherical geometry, check out Matt Zucker’s tweet. And if you want to try to draw some of these patterns for yourself, take a look at @Ayliean’s Truchet Tiles Zine.