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Let $a +_K b$ denote Knuth addition. It is easy to check that $a +_K b \equiv a + b$ mod $2$ (in fact mod $4$), so the odd numbers are Knuth-sum-free. On the other hand, note that if $a +_K b = a +_K …

This topic is fascinating, but the two specific questions asked are trivial, as indicated in the comment of Peter Taylor. Here is a paraphrase of that comment. The function $W$ has the form $W(a, b) = …

Define $f_i(n) = (-1)^{b_i(n)}$ where $b_i(n)$ is the $i$th binary digit of $n$ (Speyer's example with $\lambda = e_i$). Then $(f_1, f_2, \dots)$ defines an "infinite Hadamard matrix" because the part …

Observe that, in isolation, (the indicator function of) an arithmetic progression of common difference $2$ and length $N > n$ maps to an arithmetic progression of common difference $2$ and length $N-n …

Repeating from the comments section: This (natural and beautiful) question was previously asked and answered on this site. See Collapsible group words. It also appeared recently on math.se. The questi …

This is an answer to the follow-up question about automorphisms of a subdivision. Suppose $G$ is a connected graph which is not $2$-regular. Let $G^{(k)}$ be the $k$-subdivision of $G$, i.e., the gra …

You can take $\kappa(n) = n/2$ if $G$ is not virtually nilpotent of class $\le 2$. Let $B_n = S^{\le n}$ and $C_n = \{[b_1, b_2] : b_1, b_2 \in B_n\}$. Suppose $|C_n| < n/2$. By pigeonhole there is so …

Let $kp$ be the size of the support of $\sigma$. Let $1,2,3,4$ be four points of the ground set. The probability that $\sigma_1$ maps $1 \mapsto 2$ is $kp/n(n-1)$, because there is a $kp/n$ chance tha …

Extended comment, generalizing @IlyaBogdanov's comment about $2n-1$. Fix $n$ and let $$x_m = a(m, n) + n - m - 1.$$ Then $(x_m)$ obeys the similar recurrence $$ x_m = x_{m-1} + \gcd(x_{m-1}, n-m) - …

Yes your claim is correct for trees. Here is a standard fact about automorphism groups of trees: Lemma: If $T$ is a finite tree then there is either a vertex or an edge fixed by every automorphism of …

Just playing around with it: The RHS multiplied by $n$ is the same as $$2 \sum_{k=0}^{n-1} \binom{n+1}{k} \binom{n}{k} \binom{n+1}{k+2}.$$ Subtracting this from $n$ times the LHS gives $$\sum_{k=0}^{n …

Claim 1: (same as my comment) Let $q_1, \dots, q_k > 1$ be prime powers. Then $G = C_{q_1} \times \cdots \times C_{q_k}$ is isomorphic to a subgroup of $S_n$ if and only if $q_1 + \cdots + q_k \leq n$ …

I have come across some references for this problem, which amount to a solution very different from the (beautiful) one given by Peter Mueller, and which also goes further. It turns out that it is pos …

The first two rows are identical with probability $2^{-n}$, so $\mathbb{P}(\det M_n = 0) \geq 2^{-n}$. Incidentally, there are $2 \times \binom{n}{2}$ events like this to consider, though not quite in …

Well, here's a combinatorial proof of the identity anyway (but not a direct proof of independence of $\ell$). Write $N = R - t$. Then the identity is $$\sum_{n=0}^N \binom{n + \ell}{n} \binom{R - n-\e …