What is the proper notation for integer polynomials: $\Bbb Y=\{p\in\Bbb Q[x]\mid p:\Bbb Z\to \Bbb Z\}$?

These are known as integer-valued polynomials.. It is a classical result of Polya and Ostrowski (1920) that any integer valued polynomial, i.e. any $\,f(x)\in \mathbb Q[x]\,$ with $\,f(\mathbb Z)\subset \mathbb Z,\,$ is an integral linear combination of binomial coefficients $\,{x \choose k},\,$ see for example Polya And Szego, Problems and theorems in analysis, vol II, Problem 85 p. 129 and its solution on p. 320, or see this answer.

These results have been extended from $\,\Bbb Z\,$ to much more general rings (e.g. Dedekind domains) by Cahen at al. I don't believe that there is any standard notation to denote such rings, though I recall that some ring-theorists use the notation $\,{\rm Int}(D)$ or something similar. A search on "integer-valued polynomials" should locate much interesting literature. This paper is one convenient place to start at: J. L. Chabert, $ $ An overview of some recent developments on integer-valued polynomials. 2010.