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I am recently doing some research related to the hook length formula. The hook formula counts the number of Young tableaux of certain type.

I find there are plenty of research already been done and I can observe a lot extension and generalization of hook length formula types. But is there a good survey on those research and open problems?

What I am interested are those extensions purely combinatorial, but it is still good if anybody can provide some survey on the representation theory side.

Please correct me if there are easy-to-find survey or the problem is too general for asking.

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    $\begingroup$ Do you already know about Bruce Sagan's book The Symmetric Group? $\endgroup$
    – Nate Eldredge
    Commented Mar 31, 2015 at 21:34
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    $\begingroup$ There's a triad of papers -- arxiv.org/abs/1006.0043 , arxiv.org/abs/1006.1865 , arxiv.org/abs/1006.4593 -- which seems to cover a lot of ground (no, I have not read them). Two other approaches appear in math.ucla.edu/~pak/papers/hl7.ps and web.mit.edu/~shopkins/docs/rsk.pdf . I also remember having seen a review paper appear on the arXiv about counting Young tableaux (or P-partitions? or increasing maps between posets?) which includes the hook length formula and lots of other results (but mostly without proof). $\endgroup$
    – darij grinberg
    Commented Mar 31, 2015 at 21:53
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    $\begingroup$ a good 2-page intro with references to books is here: ams.org/notices/200702/whatis-yong.pdf $\endgroup$
    – Dima Pasechnik
    Commented Mar 31, 2015 at 22:24
  • $\begingroup$ I believe C. Krattenthaler has some nice proofs of the classical formulas. There is also the notion of hook-formulas for counting linear extensions of certain posets, (Forests, D-complete posets). See the work of Proctor for the latter, unc.edu/math/Faculty/rap $\endgroup$
    – Per Alexandersson
    Commented Mar 31, 2015 at 23:58
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    $\begingroup$ Of course the standard reference would be Chapter 7 of Stanley's EC2. It does not spend too much time on the hook-length formula but everyone interested in this kind of combinatorics should read this chapter anyways. $\endgroup$
    – Sam Hopkins
    Commented Jun 29, 2016 at 19:06

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Some references you might find interesting:

Proctor classify certain posets (d-complete posets), that admit hook formulas. There are hook formulas for forests, as well as some other types. (Victor Reiner, P-partitions revisited, Triangle Lectures in Combinatorics slides, 2011.)

Also, I believe there is some recent results on hook formulas for skew shapes. (Morales, Pak, Panova, Hook formulas for skew shapes I, 2015.)

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