How can I reorder the factors in the terms of a polynomial?

Question

How can I reorder the factors in the terms of a polynomial? Consider

poly1 = \!\(\*SubsuperscriptBox[\(x\), \(1\), \(3\)] + \*SubsuperscriptBox[\(x\), \(2\), \(3\)] + \*SubsuperscriptBox[\(x\), \(3\), \(3\)]\  - \ \(TraditionalForm\`\*SuperscriptBox[SubscriptBox[\(σ\), \(1\)], \(3\)]\)\) /.    Subscript[σ,     1] -> (Subscript[x, 1] + Subscript[x, 2] + Subscript[x, 3]) //   Expand

$$\begin{align*}-3 x_2 x_1^2-3 x_3 x_1^2-3 x_2^2 x_1-3 x_3^2 x_1-6 x_2 x_3 x_1-3 x_2 x_3^2-3 x_2^2 x_3\tag{1}\end{align*}$$

MonomialList[poly1, {Subscript[x, 1], Subscript[x, 2], Subscript[x,   3]}, "Lexicographic"]

$$\begin{align*}\left\{-3 x_1^2 x_2,-3 x_1^2 x_3,-3 x_1 x_2^2,-6 x_1 x_2 x_3,-3 x_1 x_3^2,-3 x_2^2 x_3,-3 x_2 x_3^2\right\}\tag{2}\end{align*}$$

% /. List -> Plus

$$\begin{align*}-3 x_2 x_1^2-3 x_3 x_1^2-3 x_2^2 x_1-3 x_3^2 x_1-6 x_2 x_3 x_1-3 x_2 x_3^2-3 x_2^2 x_3\tag{3}\end{align*}$$

Question1: How can I get

$$\begin{align*}-3 x_1^2 x_2-3 x_1^2 x_3-3 x_1 x_2^2-6 x_1 x_2 x_3-3 x_1 x_3^2-3 x_2^2 x_3-3 x_2 x_3^2\tag{4}\end{align*}$$

Question2: Even better if possible, how can I get

$$\begin{align*}-3\left( x_1^2 x_2+x_2^2x_1+\text{...}\right)-6 x_1 x_2 x_3\tag{5}\end{align*}$$

My goal is to keep the order of the terms in (2) unchanged when I copy (4)/(5) into an inline formula cell.

Answer 1

Question 1

As I said in a comment,

Row@MonomialList[poly1,
      {Subscript[x, 1], Subscript[x, 2], Subscript[x, 3]}, "Lexicographic"]

---

However, it works easily here because all the coefficients are negative. The following polyForm handles level 1 terms (ignores nesting). It formats the polynomial f with the monomials in a specified order. It will expand a polynomial f (via MonomialList), unless an explicit list of terms is given. It takes the same arguments as MonomialList in the case where the polynomial is passed. If the terms are passed directly in a list, they will be formatted in the same order as they occur in the list.

polyForm[f_Plus, vars_: Automatic, order_: "Lexicographic"] := 
  polyForm[MonomialList[f, vars /. Automatic :> Variables[f], order]];
polyForm[terms_List] := Module[{signs},
   signs = 
    First[Cases[#, c_?NumericQ :> Sign[c]] /. {} -> {1}] & /@ 
     Rest@terms;
   PrecedenceForm[Row[Riffle[{1}~Join~signs terms,
      signs /. {1 -> "\[MediumSpace]+\[MediumSpace]", -1 -> 
         "\[MediumSpace]-\[MediumSpace]"}]], 10]
   ];

polyForm[x_, ___] := x; (* leave other stuff alone *)

Question 2

In this case you can gather the terms by the number of variables each contains:

Total[Factor /@ Plus @@@ GatherBy[List @@ poly1, Length@Variables[#] &]]

Or you can use Row to keep them in the desired order (as in the question):

Row[Factor /@ Plus @@@ GatherBy[List @@ poly1, Length@Variables[#] &]]

Or use polyForm, which will yield the standard spacing around the last minus sign:

polyForm[Factor /@ Plus @@@ GatherBy[List @@ poly1, Length@Variables[#] &]]

---

Mr.Wizard pointed out that my order is not the same as the desired order in the OP, something I overlooked. Here is a fix:

Factor /@ Plus @@@ GatherBy[MonomialList[poly1, Variables[poly1], "Lexicographic"], 
     Length@Variables[#] &] /. poly_Plus :> polyForm[poly] // polyForm

Here is another:

MapAll[# /. poly_Plus :> polyForm[poly] &, 
 Total[Factor /@ Plus @@@ GatherBy[MonomialList[poly1, Variables[poly1], "Lexicographic"], 
     Length @ Variables[#] &]]]

For the curious, I'll share the following. Using ReplaceAll with poly_Plus :> polyForm[poly] does not work on the Total in the preceding example because is it applies the replacement at the top level first and polyForm expands the polynomial. Here is a way around that, but it does reverse the order at the top level:

Total[Factor /@ Plus @@@ 
    GatherBy[MonomialList[poly1, Variables[poly1], "Lexicographic"], 
     Length@Variables[#] &]] /. Plus -> (polyForm[List[##]] &)

Answer 2

Related questions:

• Displaying a series obtained by evaluating a Taylor series
• How to keep Collect[] result in order?
• How do I get a two-term polynomial with a leading negative sign to display in the correct (i.e. textbook) order?

As shown in answer to the third question above you could use:

HoldForm[+##] & @@ MonomialList[poly1]

(+## is shorthand for Plus[##])

As already noted above and in that answer Row is not sufficient as it will not handle the signs of terms correctly.

For the second question I am borrowing part of Michael's answer, but my result is different and I believe correct:

format = HoldForm[+##] & @@ MonomialList@ # &;

Factor /@ Plus @@@ GatherBy[List @@ poly1, Length@Variables[#] &];

Plus @@ MapAt[format, %, {1, 2}]

The order of these terms appears to match your requested output whereas Michael's output does not.

Answer 3

This can easily be done by straight manipulation with the Presentations Application, which I sell.

<< Presentations` 

step1 = poly1 // MapLevelParts[FactorOut[-3], {{1, 2, 3, 5, 6, 7}}] 
step2 = MapAt[HoldOrderForm[{1, 3, 2, 5, 4, 6}], step1, {{2, 2}}]

FactorOut will factor an arbitrary expression out of an existing expression - even if it is not initially in the expression. MapLevelParts will map a function to a set of level parts in an expression (usually a sum or product or sometimes a list). It applies as a whole to the sub sum, product or list. In your example the level parts are at level 1 but it is also possible to specify a set of level parts at a deeper position.