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Nov 13, 2014 at 3:21 comment added Timothy Chow As you probably know, this is a special case of the problem of counting the number of linear extensions of a poset. If there is no nice recurrence, you might want to check out John Stembridge's poset package for Maple, which will at least let you numerically compute these numbers. More generally, it will compute the W polynomial. If you're really lucky, perhaps the W polynomials will exhibit some pattern that will let you see how to compute them efficiently.
Nov 12, 2014 at 22:16 answer added Scott Andrews timeline score: 1
Nov 12, 2014 at 20:07 history edited Alex R. CC BY-SA 3.0
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Nov 12, 2014 at 19:52 history edited Alex R. CC BY-SA 3.0
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Nov 12, 2014 at 19:46 history edited Alex R. CC BY-SA 3.0
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Nov 12, 2014 at 19:40 history asked Alex R. CC BY-SA 3.0