The number of Alternating Sign Matrices of size $n$ is well known to be $\prod_{k=0}^{n-1}\frac{(3k+1)!}{(n+k)!}$. Is it known whether the naive q-analog expression $$\prod_{k=0}^{n-1}\frac{[3k+1]_q!}{[n+k]_q!}$$ is a polynomial in $q$ with positive coefficients? Does it come from a known statistic on ASM's?
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asked Aug 22, 2016 at 0:47
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1$\begingroup$ If I computed right then at least for $n \leq 20$ it's a polynomial in $n$ with coefficients that are not just positive but unimodal. For example, $n=7$ gives $[1, 6, 20, 49, 98, 169, 259, 359, 455, 531, 573, 573, 531, 455, 359, 259, 169, 98, 49, 20, 6, 1]$. $\endgroup$– Noam D. ElkiesCommented Aug 22, 2016 at 1:19
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1$\begingroup$ I do not get unimodality: n=4 gives for example [1, 0, 1, 1, 2, 2, 3, 2, 4, 3, 4, 3, 4, 2, 3, 2, 2, 1, 1, 0, 1] $\endgroup$– Per AlexanderssonCommented Aug 22, 2016 at 2:49
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$\begingroup$ Can someone please provide the q-expressions for n=1..4? $\endgroup$– Per AlexanderssonCommented Aug 22, 2016 at 2:54
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1$\begingroup$ My computation for n=4 agrees with the one by @PerAlexandersson. $\endgroup$– Christian StumpCommented Aug 22, 2016 at 11:45
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$\begingroup$ @NoamD.Elkies: I'm a bit puzzled, I get something very different for $n=7$: a polynomial of degree 112 with leading coefficients $1,0,1,1,2,3,5,\dots$. For $n=4$ I have precisely what Per Alexandersson has. $\endgroup$– Martin RubeyCommented Aug 23, 2016 at 11:42
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3 Answers
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It's true these polynomials are not unimodal for n=2 and up, since they all start with coefficient sequence 1 0 1 ... (The reason for this is clear from the definition of descending plane partitions.) For n = 4, 5, 6, they are also non-unimodal in the middle. But the sequence of nonzero coefficients seems to be unimodal for n=7 and above (I've calculated up to n=35). I had observed the non-unimodality of n=4,5,6 several years ago, but had thought it would continue to fail, so it's interesting to me that it seems to be (nearly-)unimodal at n=7 and beyond.
When the ASM is a permutation matrix, the right statistic is a weighted inversion count: weight each inversion pair by the larger number in the pair and then add up all the contributions. (See St000616 on FindStat.) The generating function for this statistic on permutations is an interesting analogue of the q-factorial. (See https://arxiv.org/pdf/1002.3391v2.pdf Corollary 6.)
I also think looking at the Gog or Magog triangles to try and find the right statistic is a good idea, but it's not one of the usual statistics, as far as I can tell. I've thought about this question quite a lot and would be very interested if anyone is able to make progress on finding this statistic on ASM or TSSCPP.
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$\begingroup$ Hi Jessica, welcome to MathOverflow! As a tangent, perhaps there is a nice generalization of inv and maj to ASMs, that are equidistributed, and thus give an analogue of Mahonian-ness for ASMs. $\endgroup$ Commented Aug 22, 2016 at 15:31
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1$\begingroup$ Hi @Per, yes that would be quite nice. There is a generalization of inv to ASMs; the trouble is there are some choices to make and it's not known what the right choice is. (See St000332 and St000067). You can come up with a generalization of maj too St000227, but it's not equidistributed with either of the previous statistics, so none of these statistics seem to be exactly what you would want. $\endgroup$ Commented Aug 23, 2016 at 17:45
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$\begingroup$ Mapping to SSYTs gives the possibility to compute charge, which is Mahonian on permutations. Perhaps that could be something.. $\endgroup$ Commented Aug 23, 2016 at 17:50
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The naive $q$-analog of that expression is naturally a generating function for descending plane partitions by weight (not sure of reference, though), but it doesn't translate to any natural statistic on ASMs/monotone triangles/etc., and it's not known what statistic on ASMs does the trick.
In terms of unimodality, I would imagine there's a way to prove it algebraically by induction. There's a trick along the lines of 'If $f(q)$ is unimodal, and $f(q)/[n]_q$ is a polynomial, then $f(q)/[n]_q$ is also unimodal.' that might work.
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Perhaps it is more natural to look at Gog or Magog-triangles, which are special types of GT-patterns, equinumerous with ASMs.
These GT-patterns can then be mapped to SSYTs of triangle shape, and on these, one can do lots of statistics.
I tried a few, but no obvious candidate.
EDIT:
This questing has been asked in the end of this article (notices of the AMS). There seem to be a cyclic sieving phenomenon on ASM, but they write that they have no idea what the q-statistic is. However, the CSP might give a lead.
answered Aug 22, 2016 at 2:53