Yet Another Expression Ordering Question (not polynomial but transcendental)

Question

Forcing a particular ordering of terms, i.e. overriding Mathematica's, is a recurrent issue, especially for users working within particular domains where there is a particular ordering tradition or where the problem itself is more "elegant" to a user when expressed in a certain way.

Both @Silvia and @Mr Wizard's answers to How do I reassign canonical ordering of symbols? are effective and educational (but see also 188576, 30216 and tag combination [formatting] [order]) but I have failed to apply the answers to this problem.

The issues involve the facts that the expression of interest is not a polynomial - MonomialList does not apply (though one can of course use List@@colEqn to get the terms of the Plus[]) - and that ordering information is implicit and somewhat buried.

The context is the Collatz problem (an entertaining diversion, no more than that!) in which, if one redefines "step" to be $n\to n'=\frac{3 k+1}{2^q}$ s.t. $n'$ is again odd, one can write the nth term in the sequence starting from an initial value k to be, for presentation purposes:

colEqnH = 
 HoldForm[(3^p  * 
      k + (3^(p - 1)) + (3^(p - 2)) 2^Subscript[q, 
       1] + (3^(p - 3)) 2^(Subscript[q, 1] + Subscript[q, 
         2])  + \[Ellipsis]  + (3^1) 2^(Subscript[q, 1] + Subscript[q,
          2] + \[Ellipsis]  + Subscript[q, n - 2]) + 
     2^(Subscript[q, 1] + Subscript[q, 2] + \[Ellipsis]  + Subscript[
        q, n - 1]))/2^Subscript[q, n]]

$\frac{3^p k+3^{p-1}+3^{p-2} 2^{q_1}+3^{p-3} 2^{q_1+q_2}+\ldots +3^1 2^{q_1+q_2+\ldots +q_{n-2}}+2^{q_1+q_2+\ldots +q_{n-1}}}{2^{q_n}}$

But on releasing the hold the output becomes:

$2^{-q_n} \left(k 3^p+3\ 2^{q_{n-2}+q_1+q_2+\ldots }+2^{q_{n-1}+q_1+q_2+\ldots }+3^{p-3} 2^{q_1+q_2}+3^{p-2} 2^{q_1}+3^{p-1}+\ldots \right)$

What I would like is to be able to manipulate an unheld equation and then to be able to present it in the preferred form.

Features of note/requirements for formatted output

Powers of 3: the terms should be left to right in decreasing order of the powers of 3 - but the order is only implied by $p > p-1 > p-2 > ... > 1$
In products, powers of 3 come first (before k, powers of 2)
Powers of 2: the terms should be left to right in order or the subscripts (indices) of the $q$
Ellipsis is used to indicate the omission of an indefinite number of terms, and its position in the input expression should be preserved
superscripts/subscripts/exponents should be written $p-1$ and not $-1 +p$
Fractions should be displayed as $\frac{a}{b}$

For reference, I generalised @Silvia's code and applied it as follows (clearly without success, but.. it shows effort :) )

ClearAll[prodSort]
prodSort[mono_, prodPriorityTable_] :=
    If[Head[mono] =!= Times,
        Row[{mono}],
        Module[{termLst},
            termLst = List @@ mono;
            Row@SortBy[{termLst, prodPriorityTable@termLst}\[Transpose], Last][[All, 1]]
        ]
    ]
ClearAll[sumSort]
sumSort[polynom_, sumPriorityTable_] :=
    Module[{termLst},
        termLst = List @@ polynom;
        SortBy[{termLst, sumPriorityTable@termLst}[Transpose], Last][[All, 1]] // Riffle[#, "+"] & // Row
    ]
containsQ[expr_, form_]:=Not@FreeQ[expr, form];
ClearAll[collatzProdPriorityTable];
collatzProdPriorityTable[termLst_]:= 
    Piecewise[
        {
            {  0, containsQ[#, Power[3,_]]},
            { 20, containsQ[#, Power[2,_]]},
            {100, containsQ[#, n]}
        }, 50
    ] & /@ termLst;
ClearAll[collatzSumPriorityTable];
collatzSumPriorityTable[termLst_]:=
    Piecewise[
        {
            { 0, containsQ[#, n]},
            {10, containsQ[#, 3]}
        }, 50
    ] & /@ termLst;
present[expr_, sumPriorities_, prodPriorities_]:= sumSort[prodSort[#, prodPriorities] & /@ expr, sumPriorities] // TraditionalForm

And finally, the imperfect result from:

present[ReleaseHold@colEqnH, collatzSumPriorityTable, collatzProdPriorityTable]

No TeX available because "TeXForm of TemplateSlotSequence[1, ] is not supported"; 55363 applies? (I am using MMA 11.0.1) And yes, I now see I haven't handled the lastly-introduced fraction properly, but that doesn't affect the thrust of the question)

UPDATE

Using @kglr's tip setting BoxForm`$UseTemplateSlotSequenceForRow = False; I was able to obtain TeX at last

$2^{-q_n}+k 3^p+3\ 2^{q_{n-2}+q_1+q_2+\ldots }+2^{q_{n-1}+q_1+q_2+\ldots }+3^{p-3} 2^{q_1+q_2}+3^{p-2} 2^{q_1}+3^{p-1}+\ldots$

Yet Another Expression Ordering Question (not polynomial but transcendental)

0 Answers0