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I am glad to accept your paper.
Both reviewers consider that the manuscript is ready for publication. Reviewer 1 detected some typos and inconsistencies that should be easy to address.
The article was substantially improved from the first version. However I still have a few suggestions to improve its readability.
something went wrong on page 5, line 140: "For intermediate tucks, and to avoid..." the symbols for Tie3 are wrong.
The labels used in the various grammars are inconsistent. While this does not invalidate the results, it makes it substantially more difficult to understand the role of each rule. For instance, I do not understand the logic in introducing <last_> in the first grammar on page 6, while this was not present in the first grammar in Section 4.1. (I think it is more clear with "last" present)
In the grammar given in 4.2, <label> can refer to either information at the current position (for instance <tw2>: "A this stage the surrounding context has two more T than W mod 3") or information about the content (for instance <wstart>: this is a tuck starting with W). This is very annoying and I think unneccessarily complicates the grammar and misses the opportunity of fully using the power of Theorem 1. I suggest:
<wstart> ---> <wtuck2> (the 2 hints that #W-#T = 2 mod 3 inside)
<tstart> ---> <ttuck1> (the 1 hints that #W-#T= 1 mod 3 inside)
<tw?> and <wt?> both to <wX>, which indicates that #W - #T mod 3 = X.
In this way, with minor adjustments, one needs only 3 rules <wX> instead of 6 <tw?> or <wt?>, one only ever considers #W-#T (not #T-#W, which only complicates things), and the grammar of 4.2 only ever uses labels that refer to their content, not the position (which is a legacy from the original tie problem, which Theorem 1 dispenses us of). Simplifying the grammar in this way might also be helpful for the generating function that memory-bombed later in the paper.
When defining the grammars, particularly in 4.2, a reminder on the precedence of | would be useful. More spacing would also be helpful.
Theorem 1 should be rephrased: "if the position is preceded by..." "If the subsequence of the last 2k W or T symbols is such that"
The automaton on page 7 and its description don't match: "Execution starts in the middle node" ???
Page 7, line 179 "At most context-free" -> "At worst context-free". This introduces a value judgement (more properties == better), but what does it really mean if left as is?
Page 10, line 249: Unclear to me what U'UU is.
The counts are now dependent on the correct implementation of a Maple package, and the correct entering by the authors of the grammar they described.
I didn't check either.
A nice report, with a clear narrative
The grammar describes well the problem, and the generating function resolution is elegant.
The mathematics appears to be correct.
Kudos on the generating functions. It is a much improved paper, with a much clearer focus. I recommend that you follow up further with Flajolet and Sedgewick's "Analytic Combinatorics" Cambridge UP 2009 for more about what you can do with these generating functions.
Both reviewers agree that the paper is valid and interesting, and both have constructive suggestions to enhance and tighten the presentation. The authors should first address issues common to both reviews: the uneasy feelings arising from the excerpt on page 1, the need for more figures and/or photos, and the relevance of the long annexed table. I also look forward to comments on the scientific and methodological questions raised by the reviewers.
Concerning the Figshare issue discussed by Reviewer 1, links to external sites are perfectly acceptable, as long as they remain accessible.
Fink and Mao described a formalism to enumerate usual tie knots (four-in-hand or Windsor for instance). One contribution of this paper is to simplify the original formalism.
One of the characters in the movie "The Matrix Reloaded" used a tie knot of a distinctively different style, wrapping the thinner blade around the other. This leads to tie knots whose "facade" is not plain, since it shows more of the internal structure of the knot. The natural question is then to enumerate tie knots following this new pattern, which is the main realization of this paper. This result has already received a lot of attention in mainstream media.
The enumeration relies on a presentation through formal languages, and qualifies that language (regular, context sensitive) in the different subcases considered.
The results are valid, and the conclusions are clearly stated. The paper includes (long) tables of knots satisfying different constraints and a printout of the code used to produce these tables. This code has been put on Figshare, to ensure complete transparency. I do not know if PeerJ has a policy on the topic, but it would seem to me desirable to keep both code and paper tied for archival purposes.
This paper looks like a worthwhile opportunity to include a closeup picture of a real knot. This would dramatically increase its accessibility.
p. 1 l. 18: The story presented here comes with no context. Who are those people, where did this take place? Knorr tells the story on his blog, but it is not clear where the original conversation took place.
p. 3, l. 95-96 "No region/direction shall repeat": unclear phrasing
p.3 l. 96: It would be helpful to have a reminder somewhere, maybe here, of which symbol is inwards, which is outwards.
p.4 l. 123: As far as I can tell, theorem 1 does not exclude tucks that would be simultaneously k-fold or k'-fold (from just reading the sequence of windings). Is this correct? If so, does your enumeration distinguish this? Please clarify. In the cases where there are multiple tucks, you should probably say why it is not a concern that a tuck makes later tucks spanning more windings impossible (if I have two overlapping bows, and tuck under the top one, it makes it impossible later to tuck under the bottom one).
p. 5 The rules after line 156 and line 157 are contradictory (in presentation only)
p. 6 l. 171: I think this automaton is wrong. It allows for the following sequence "TTTUTTTUTTTU", which should not be allowed according to 4.1
p. 7 line 206: finish the sentence with "... k-2 W and T symbols with proper counts".
I have labeled the article "Major revision" because I want to make sure the language used to make the corrections is clear enough (i.e. that this goes through me again). The changes asked are not actually major.
The problem is novel, and the authors touch on several interesting aspects. There should be more useful figures.
The authors describe a novel formal language which models "necktie configurations". They propose a modification of classic work, which eliminates and ending configuration. The language is given explicitly a as an attribute grammar.
The formulas appear to be correct.
As a matter of form, I would add that the first page is dominated by a verbatim history of the problem. It is unusual to me to see such a large amount of "fact" text copied from another source. Not all of this information is necessary, and I recommend the authors edit this section considerably.
The proofs of theorems 3 and 4 are adhoc combinatorial arguments, but could be harmonized if more sophisticated enumeration tools were used. There are several formalisms which translate grammars into enumerative generating functions, and it seems like these should be a good candidate for such a process. Hence, from an enumeration point of view, it is lacking as the state of the art is more sophisticated. At the very least, they should offer an asymptotic comparison.
From a language theoretic point of view it is interesting that it is a context-sensitive language. Is there a similar argument for the Fink and Mao case?
If the intended goal is to make new, usable patterns for ties, then the authors should add some focus on this-- can they highlight some new knots that one could actually wear, (for which they provided have criteria).
The full tables at the back seem unnecessary given that they describe how to generate them.
The authors definitely succeed on the whimsy front, and it was enjoyable to read. From a combinatorics point of view, it is a relatively minor adaptation from the existing case and as such could benefit from a tighter presentation, or a closer investigation into modern enumeration techniques.
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