Collecting multivariate polynomials

Question

In a way of minimal example, say I want to collect the expansion of $(a+b+x+y)^2$ with respect to $x$ and $y$. Collect[(a+b+x+y)^2,{x,y}] gives nested collection (sums involving $y$ as coefficients at powers of $x$) which is not what I want. Most concise code that I could find for it is

Total[Map[Last[#]Times@@({x,y}^First[#])&,CoefficientRules[(a+b+x+y)^2,{x,y}]]]

(which correctly gives a^2 + 2 a b + b^2 + (2 a + 2 b) x + x^2 + (2 a + 2 b) y + 2 x y + y^2).

Is there more straightforward way to do it?

Answer 1

One can use MonomialList to separate power products involving a specified set of variables. In this example it might be done as below.

Plus @@ MonomialList[Expand[(a + b + x + y)^2], {x, y}]

(* Out[1549]= a^2 + 2 a b + b^2 + (2 a + 2 b) x + x^2 + (2 a + 2 b) y + 
 2 x y + y^2 *)

Answer 2

Try this:

lst = Rest @ Flatten @ Array[x^# y^#2 &, {3, 3}, 0]


(* {y, y^2, x, x y, x y^2, x^2, x^2 y, x^2 y^2} *)

and then

 Collect[Expand[(a + b + x + y)^2], lst]

(*  a^2 + 2 a b + b^2 + (2 a + 2 b) x + x^2 + (2 a + 2 b) y + 2 x y + y^2  *)

Have fun!