# More on rearrangements of series . . .

Earlier I posted this question. First I'll quote the question before refining it and elaborating on it:

For which classes $C$ of bijections from $\{1,2,3,\ldots\}$ to itself is it the case that for all sequences $\{a_i\}_{i=1}^\infty$ of real numbers, if $\displaystyle f\mapsto \lim_{n \to\infty} \sum_{i=1}^n a_{f(i)}$ remains the same for all $f\in C$ then it remains the same for all bijections $f$ from $\{1,2,3,\ldots\}$ to itself?

Thus if you have a conditionally convergent series, then at least one of the rearrangements that change the value of the sum is within $C$.

Among the replies, two concrete examples were given by Robert Israel and Aaron Meyerowitz:

• For example, it is true if $C$ contains a representative of every equivalence class of bijections of $\mathbb N$, where $f \sim g$ if there exist $N,M$ such that $|f(i) - g(i)| < M$ for all $i > N$. (Robert Israel)

• It is true if $C$ is the set of involutions $(i_1j_1)(i_2j_2)\cdots$ or even the set of all involutions with the added condition $$i_1 \lt i_2 \lt \cdots\text{ and }j_1 \lt j_2 \lt \cdots \tag{*}$$ (Aaron Meyerowitz)

To these I will add two more examples before getting to the actual question:

• Suppose the sequence $\big((m_i,n_i) : i=1,2,3,\ldots\big)$ lists every pair of positive integers exactly once. Suppose that for every permutation $\sigma$ of $\{1,2,3,\ldots\}$ we have $$\lim_{N\to\infty} \sum_{i=1}^N a_{m_i,n_i} = \lim_{N\to\infty} \sum_{i=1}^N a_{m_{\sigma(i)},n_{\sigma(i)}}.$$ Then the sum is also unaltered by the "rearrangement" $$\sum_{m=1}^\infty \left( \sum_{n=1}^\infty a_{m,n} \right),$$ in which the infinitely many terms $\sum_{n=1}^\infty a_{1,n}$ come before the infinitely many terms $\sum_{n=1}^\infty a_{2,n},$ which in term precede infinitely many other terms, etc., and by a great variety of other rearrangements.

• Let $$e_{n,k} = \sum_{\begin{smallmatrix} I\,\subseteq\,\{1,\,\ldots\,,n\} \\ |I|\,=\,k \end{smallmatrix}} \prod_{i\,\in\,I} x_i$$ be the $k$th-degree elementary symmetric polynomial in variables $x_1,x_2,x_3,\ldots.$ (So for example, $e_{4,2} = x_1x_2 + x_1x_3 + x_1 x_4 + x_2 x_3 + x_2 x_4 + x_3 x_4,$ and $e_{n,0}=1$ and $e_{n,k}=0$ if $k>n.$) Further, let $$e_n = \sum_{\begin{smallmatrix} I\,\subseteq\,\{1,2,3,\,\ldots\,\} \\ |I|\,=\,k \end{smallmatrix}} \prod_{i\,\in\,I} x_i.$$ Suppose that the sum of the infinite series $$e_0-e_2+e_4-e_6+\cdots,$$ (each term of which is an infinite series in its own right) regardless of the order of the terms, is unchanged by any rearrangement that results from permutation of the variables $x_1,x_2,x_2,\ldots\,.$ Then that sum is also unchanged by all other rearrangements, i.e. those that do not result from permutation of the $x$s. (At the time I posted my earlier question, I wondered if I could prove this by some method that does not speak of the sum of absolute values being finite. I still don't know whether that can be done. I proved it by conventional and routine methods involving absolute values.)

In my earlier question, the body of the question itself said nothing about cardinality, but of course I had to write a subject line, and I wrote:

How many rearrangements must fail to alter the value of a sum before you conclude that none do?

In a sense I might be said to have intended the phrase "how many" to be taken at least partially literally. But not entirely.

I "accepted" the answer from Joel David Hamkins, which in effect took those words literally and answered the question that way. In effect, he asked how small the cardinality of such a class can be.

SO MY PRESENT QUESTION attempts to narrow the meaning and broaden the list of examples, and ask this: Might there be any nice general descriptions of classes $C$ of the kind referred to in the words $\text{“}$For which classes $C\ldots\text{?''}$ above?

• Isn't the problem to forget about convergent series and really look at $Sym(\mathbb{N})/ \sim \ \$ where $\sim$ is the equivalence relation defined by Robert Israel ? – reuns Jun 17 '17 at 21:32
• @reuns : ok, I'm thinking about what your question means. In the meantime, do you see the difference between the following, and the difference betwen their MathJax coding, and why someone might consider the former incorrect?: \begin{align} & \operatorname{Sym}(\mathbb N)/\sim \\ \\ & \operatorname{Sym}(\mathbb N)/{\sim} \end{align} – Michael Hardy Jun 17 '17 at 21:36
• @reuns : Can you elaborate a bit? Why would that answer this? – Michael Hardy Jun 17 '17 at 21:37
• Clearly if $l = \sum_n a_n$ then for any $\sigma \sim Id$ : $l = \sum_n a_{\sigma(n)}$. Intuitively, the converse should hold : that for any $\sigma \not\sim Id$ we can find a series such that $l = \sum_n a_n \ne \sum_n a_{\sigma(n)}$. In that case you really want to find an explicit description of $\text{Sym}(\mathbb{N})/ \sim$ (and you can forget about conditionally convergent series) – reuns Jun 17 '17 at 22:04
• For reference, our paper (with the OP as co-author) is available at The rearrangement number (arxiv:1612.07830), and see also my introductory talk. The paper provides various sufficient criteria for a family to have the rearrangement property, such as theorem 11: every co-meager family of permutations successfully tests for absolute convergence. Necessary criteria arise from the lower bounds on the rearrangment number. – Joel David Hamkins Jun 18 '17 at 2:14