Is there a common symbol for concatenating two (finite) sequences? Say we have two finite sequences $X = (x_0,...,x_n)$ and $Y = (y_0,...,y_n)$.
Is there a more or less common notation for the concatenation of these sequences, like $\sum (X,Y) = (x_0,...,x_n,y_0,...,y_n)$?
 A: Computer science often uses the ⧺ (U+29FA) symbol for concatenation.
This is \doubleplus in the LaTeX package unicode-math (which requires a modern engine that supports Unicode), as well as the legacy packages stix and stix2. Or, in the modern toolchain, you can use the Unicode symbol in your source.
A: If $x$ and $y$ are finite sequences, you could denote their concatenation by $xy$. Let me explain. There's at least two ways of formalizing the statement "$x$ and $y$ are finite sequences in $X$"


*

*$x$ and $y$ are functions of type $[\:\!n) \rightarrow X$, where $[\:\!n)$ is a shorthand for the set $\{0,\ldots,n-1\}$.

*$x$ and $y$ are elements of $X^*$, where $X^*$ is the monoid freely generated by $X$. 
If you're interested in concatenating these things, then you should probably take the second perspective, in which case the concatenation of $x$ and $y$ is simply their product in the monoid $X^*$, which is denoted $xy$.
A: The comments suggest the following notations for the concatenation of $X$ and $Y$:


*

*$X^\frown Y$ (given by X^\frown Y);

*$XY$ (given by XY);

*$X \cdot Y$ (given by X \cdot Y);

*$X \mathbin\Vert Y$ (given by X \mathbin\Vert Y);


of which the first seems not to be in use for other concepts, making it especially suitable.
A: In formal specifications, one way to concatenate two sequences is using the Haskell concatenation symbol as indicated in one of the comments above. In a $\mathrm\TeX$ editor one can type the following:   X +\!\!\!+ Y. The result appears like this, $X+\!\!\!+Y$.
A: Starting with your defined sequences $X = (x_0, \ldots, x_n)$ and $Y=(y_0,\ldots,y_n)$, you can use the commonly accepted tuple/ordered pair notation:
\begin{align}
   (X,Y) &= \left(
    \left(x_0,\ldots,x_n\right),
    \left(y_0,\ldots,y_n\right)
   \right)
   \\
   &= \left(x_0,\ldots,x_n,y_0,\ldots,y_n\right)
  \end{align}
A: The same question on Tex SE.
From there, and more:


*

*$X \oplus Y$ (given by X \oplus Y);

*$(X,Y)$ (given by (X,Y));


I would avoid $X \times Y$, $XY$ or $X \cdot Y$ to not confuse it with any sort of multiplication / product.
And I would also not use $X \otimes Y$ because it is usually the tensor product. (See also here.)
Some relevant Wikipedia pages with common notations:


*

*Vector space

*Direct sum

*Tensor product
A: $\newcommand\mdoubleplus{\mathbin{+\mkern-10mu+}}$
In haskell the $ \mdoubleplus $ operator is used for concatenating lists.
You can define it in latex using the command
\newcommand\mdoubleplus{\mathbin{+\mkern-10mu+}}

A: First of all, the $\frown$ (like the $\smile$) are used in algebraic topology for the cap (cup) product, so there is another use for this symbol.  I haven't seen $\uplus$ suggested and I have searched extensively to find an existing use without success.  It seems especially suited because sequences are in fact index-ordered sets which can contain duplicate elements.  Concatenation is the union of such sets which preserves the order of concatenation and allows duplicates.
