Incompatibility of Row and TeXForm

The following works in Version 9.0.1.0 and Version 10.0.1.0

BoxForm`$UseTemplateSlotSequenceForRow = False; 
{x^a, Sqrt@b,  ArcSin[c]} // Row // TeXForm
(* x^a\sqrt{b}\sin^{-1}(c) *)

Short answer

The TLDR answer is:

System`Convert`CommonDump`templateBoxToDisplay = BoxForm`TemplateBoxToDisplayBoxes

A more long-winded answer follows.

Background

The basic approach used by TeXForm is to create TraditionalForm boxes, and then to convert those boxes into a TeXForm string. The basic internal function that TeXForm uses to convert boxes into a string is System`Convert`TeXFormDump`maketex. For example, consider BesselJ[n, x]. The TraditionalForm boxes are:

MakeBoxes[BesselJ[n, x], TraditionalForm]

TemplateBox[{"n", "x"}, "BesselJ"]

Notice how System`Convert`TeXFormDump`maketex is used to convert the above boxes into a string:

TracePrint[
    ToString[BesselJ[n, x], TeXForm],
    _System`Convert`TeXFormDump`maketex,
    TraceInternal->True
]

System`Convert`TeXFormDump`maketex[TemplateBox[{n,x},BesselJ]]

System`Convert`TeXFormDump`maketex[RowBox[{SubscriptBox[J,n],(,x,)}]]

...

"J_n(x)"

By default, System`Convert`TeXFormDump`maketex uses System`Convert`CommonDump`ExpandTemplateBoxes to convert TemplateBox objects into normal boxes:

TracePrint[
    System`Convert`TeXFormDump`maketex[TemplateBox[{"n","x"},"BesselJ"]],
    _System`Convert`CommonDump`ExpandTemplateBoxes
]

System`Convert`CommonDump`ExpandTemplateBoxes[TemplateBox[{n,x},BesselJ]]

System`Convert`CommonDump`ExpandTemplateBoxes[TemplateBox[{n,x},BesselJ],Display]

"J_n(x)"

and:

System`Convert`CommonDump`ExpandTemplateBoxes[TemplateBox[{"n","x"},"BesselJ"]]

RowBox[{SubscriptBox["J", "n"], "(", "x", ")"}]

Summarizing, TeXForm does the following:

1. Create TraditionalForm boxes
2. Convert TemplateBox objects into normal boxes
3. Convert normal boxes into a TeXForm string

The problem with Row

So, where do things go wrong with Row objects? In Mathematica version 7 and earlier, Row objects were formatted as RowBox objects directly, and so there was no need to convert TemplateBox objects into normal boxes. However, starting in versions 8 or 9, Row objects were formatted as TemplateBox objects. For example, in M9 (I don't have ready access to earlier versions):

MakeBoxes[Row[{x, y}], TraditionalForm]

TemplateBox[{"x", "y"}, "RowDefault"]

One possible fix for this issue is to restore the old functionality of formatting Row objects as normal boxes, and this is what the global variable BoxForm`$UseTemplateSlotSequenceForRow controls. I don't like this solution, as it means that Row objects will never use TemplateBox, which I think is a shame (I think TemplateBox offers both speed and memory advantages). Also, it only fixes one symptom of the issue, and not the underlying problem. So, let's figure out the underlying problem.

TemplateSlotSequence

Here is the TemplateBox format for a Row object:

boxes = MakeBoxes[Row[{x, y}, None], TraditionalForm]

TemplateBox[{"x", "y"}, "RowNoSeparators"]

(I used a Row object with a separator for reasons I will explain later). These are the normal boxes that TeXForm creates for this TemplateBox:

System`Convert`CommonDump`ExpandTemplateBoxes[
    TemplateBox[{"x","y"}, "RowNoSeparators"]
]

RowBox[{TemplateSlotSequence[1]}]

Notice the appearance of a TemplateSlotSequence object. This kind of object is not supported by System`Convert`TeXFormDump`maketex. Also, just like other Slot objects, the TemplateSlotSequence object only has meaning when it is inside of a Function object, so the above output is not a valid form of normal boxes. This is the reason for the error that gets reported:

ToString[Row[{x, y}, None], TeXForm]

TeXForm::unspt: TeXForm of TemplateSlotSequence[1] is not supported.

""

Single argument Row

The above error doesn't happen for single argument Row because System`Convert`TeXFormDump`maketex was hacked to support it. Compare:

TracePrint[
    System`Convert`TeXFormDump`maketex[TemplateBox[{"x", "y"}, "RowNoSeparators"]],
    _System`Convert`CommonDump`ExpandTemplateBoxes
]

System`Convert`CommonDump`ExpandTemplateBoxes[TemplateBox[{x,y},RowNoSeparators]]

System`Convert`CommonDump`ExpandTemplateBoxes[TemplateBox[{x,y},RowNoSeparators],Display]

TeXForm::unspt: TeXForm of TemplateSlotSequence[1] is not supported.

""

with:

TracePrint[
    System`Convert`TeXFormDump`maketex[TemplateBox[{"x", "y"}, "RowDefault"]],
    _System`Convert`CommonDump`ExpandTemplateBoxes
]

"xy"

Notice that the "RowDefault" TemplateBox never gets processed by System`Convert`CommonDump`ExpandTemplateBoxes because System`Convert`TeXFormDump`maketex has a special DownValues for this case.

Non-Row TemplateSlotSequence issues

The failure to support TemplateSlotSequence is not restricted to Row objects. For example:

ToString[BellY[a, b, {c, d}], TeXForm]

TeXForm::unspt: TeXForm of TemplateSlotSequence[3,,] is not supported.

"Y_{a,b}()"

ToString[WignerD[{j, m1, m2}, \[Theta]], TeXForm]

TeXForm::unspt: TeXForm of TemplateSlotSequence[4,,] is not supported.

"D_{\text{m1},\text{m2}}^j()"

So, changing the formatting of Row so that it doesn't use a TemplateBox is not really sufficient, it won't fix the above errors.

Fix

The function used by TeXForm to convert TemplateBox objects into normal boxes is System`Convert`CommonDump`templateBoxToDisplay:

TracePrint[
    System`Convert`CommonDump`ExpandTemplateBoxes[TemplateBox[{"x", "y"}, "RowNoSeparators"]],
    _System`Convert`CommonDump`templateBoxToDisplay,
    TraceInternal->True
]

System`Convert`CommonDump`templateBoxToDisplay[TemplateBox[{x,y},RowNoSeparators]]

RowBox[{TemplateSlotSequence[1]}]

Instead of using System`Convert`CommonDump`templateBoxToDisplay one could use BoxForm`TemplateBoxToDisplayBoxes:

template = TemplateBox[{"x","y"},"RowNoSeparators"];

System`Convert`CommonDump`templateBoxToDisplay[template]
BoxForm`TemplateBoxToDisplayBoxes[template]

RowBox[{TemplateSlotSequence[1]}]

RowBox[{"x", "y"}]

Note that the latter function produces acceptable boxes. I don't know why TeXForm uses its own custom function to convert TemplateBox objects. So, a simple fix is to redefine System`Convert`CommonDump`templateBoxToDisplay:

System`Convert`CommonDump`templateBoxToDisplay = BoxForm`TemplateBoxToDisplayBoxes;

Now, TeXForm will work properly with any object whose format uses a TemplateSlotSequence TemplateBox:

ToString[BellY[a, b, {c, d}], TeXForm] //OutputForm
ToString[WignerD[{j, m1, m2}, θ], TeXForm] //OutputForm
ToString[Row[{a, b}, ","], TeXForm] //OutputForm

$Y_{a,b}(c,d)$

$D_{\operatorname{m1},\operatorname{m2}}^j(0,\theta ,0)$

$a,b$