I can't find this question asked anywhere else, however that may be because arithmetic progressions within permutations are an existing object of study that seemed to be returned in all my searches. What I want is the opposite: an arithmetic progression whose terms are permutations of the digits of each other, for instance:
$$560, 605, 650$$
are in arithmetic progression, and are all formed by the 'concatenation of a permutation' of the digit set $\{0,5,6\}$.
More precisely, I'd like to define an arithmetic permutation progression (APP) of degree $n$ and length $m$ in base $\beta$ as a sequence $(a_i)_{i=1}^{m}$ where $a_{i+1}=a_i+k$ for $1\leq i < m$ and $a_1=N$ for some $N, k \in\mathbb{N}$, and if $N$ has base $\beta$ representation $(d_nd_{n-1}\cdots d_2d_1)_\beta$ then each $a_i=(d_{\sigma_i(n)}d_{\sigma_i(n-1)}\cdots d_{\sigma_i(2)}d_{\sigma_i(1)})_\beta$ for some permutation $\sigma_i\in S_n$.
When $m=1,2$, any singleton or pair of numbers is trivially in arithmetic progression, and so any distinct permutations form an APP. The first 'interetsing' case is $m=3$. Note that to have any APPs we need $n!\geq m$, so take $n=m=3$
I wrote some fairly crude python to find all the solutions in this case. There are $14$ sequences for $\beta=10$, but the common difference $k$ only obtains two values: $k=45$ and $k= 333$. Indeed, it seemed that $k$ only obtained at most two values in any base, and more often than not, only one.
So I did some investigation, and after several pages of algebra and consideration of many cases and sub-cases, I reached the following conclusion:
- If $\beta$ is even, then there are $\beta-2$ APPs with $k=\frac12\beta(\beta-1)$,
- If $\beta$ is odd, then there are $\beta-1$ APPs with $k=\frac12(\beta^2-1)$,
- If $\beta \equiv 1 \pmod{3}$ then there are $\frac23(\beta-1)$ APPs with $k=\frac13(\beta^3-1)$.
However, I hope that there is a more elegant and concise argument than mine, which might be more condusive to the extension to the case $n=m=4$ and beyond, e.g: is there an 'obvious' reason that $45$ and $333$ are the only possible values of $k$ in base $10$? What conclusions can be drawn for $n=m=4$? Can a general argument be made?
More Observations:
- The digit sum of the terms of an APP are clearly all the same.
- For $n=m=3$, the digit sum of $k$ is always $\beta-1$, i.e: $k$ is a mutliple of $\beta-1$.
- It would appear $\beta-1|k$ for $(n,m)=(4,3),(4,4)$ too.
- For $\beta=10$: $n=3$ has APPs with $k_3=45=9\cdot5$, $n=4$ has APPs with $k_4=495=9\cdot55$. If this pattern continues, do we have APPs in even base $\beta$ with $k_n=\frac12\beta(\beta^{n-2}-1)$? Unfortunately, my code isn't good enough to handle $n,m\geq5$.