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August 5

Easy formula for volume of the Triangular pyramid with right angle corners at top?

Is there an easy formula for the volume of a triangular pyramid where the "top" point has three right angles, the top edges are all the same and the bottom is a regular triangle? (top edges are length m, so bottom edges are m*sqrt(2) )Naraht (talk) 12:56, 5 August 2022 (UTC)[reply]

The volume of any pyramid is A*h/3 where A = area of base and h = the height of the pyramid (perpendicular distance from the apex to the plane containing the base). This works for any pyramid no matter how regular or odd shaped. --Jayron 32 13:04, 5 August 2022 (UTC)[reply]

Yes, but I'm not sure what the height is of that pyramid. I guess if you turn it over onto one of its non-regular triangle faces you can get that. A=1/2m for that, and then the height would be m, so V=1/6 m^3?Naraht (talk) 13:07, 5 August 2022 (UTC)[reply]

A = 1/2 m^2. --Lambiam 14:16, 5 August 2022 (UTC)[reply]

Yes. And this is a good way to compute the height of the pyramid with respect to the equilateral triangle. 66.44.49.56 (talk) 14:27, 8 August 2022 (UTC)[reply]

Inverse of an order matrix

Let P be a finite partially ordered set. Define a matrix M by M_ab = 1 if a ≤ b, and 0 otherwise. Is there a combinatorial interpretation of the the entries in M^-1? For example, if P is the set {1, 2, 3, 4} and the relation is divisibility, the M is:

And M^-1 is:

In this case M^-1 seems to be closely related to the Möbius function. This came up when I was (still) trying to find a formula for the number of balanced sets in an n-cube from last week's question. In that case M was the set of partitions of a finite set ordered by refinement. (Actually, it was a modification of M with rows and columns merged due to symmetry.) --RDBury (talk) 17:06, 5 August 2022 (UTC)[reply]

Yes it is always the matrix of the values of the Mobius function. — Preceding unsigned comment added by JayBeeEll (talk • contribs)

If it's if's a divisibility matrix then yes, but for an arbitrary partially ordered set then no. For example, for the partitions of a set of order 3, the matrix M is:

1  1  1  1  1
0  1  0  0  1
0  0  1  0  1
0  0  0  1  1
0  0  0  0  1

and M^-1 is:

1 -1 -1 -1  2
0  1  0  0 -1
0  0  1  0 -1
0  0  0  1 -1
0  0  0  0  1

But 2 is not a value of the Möbius function. There may be generalization of the Möbius function I'm not familiar with though. --RDBury (talk) 18:37, 9 August 2022 (UTC)[reply]

Oh I see, I didn’t even understand the confusion. It’s called the Mobius function of the poset. Apparently the relevant article on WP is incidence algebra but any decent text on posets should cover it (e.g. certainly EC1, chapter 3). — Preceding unsigned comment added by JayBeeEll (talk • contribs)

Thanks. I saw the article on incidence algebras and skimmed it, but I didn't see the relation to posets right away so I didn't know if it was worth pursuing. A more general problem is to find a combinatorial interpretation of the coefficients of M^k for any k. This is a polynomial in k, so if that's possible then it's just a matter of evaluating the polynomial at -1. M² is straightforward, just the number of elements in the interval [a, b]. RDBury (talk) 16:54, 10 August 2022 (UTC)[reply]

Powers of M count multichains. There's a well-developed theory here that sadly is woefully underrepresented on WP; I strongly recommend you consult chapter EC1 if you're interested in what is known. -- JBL (talk) 18:24, 10 August 2022 (UTC)[reply]

Powers of M counting multichains sounds right, assuming a multichain is what I think it is. I'm not sure how you'd interpret negative powers though. By EC1 I assume you mean Richard Stanley's Enumerative Combinatorics: Volume 1. Getting my hands on a copy is a bit inconvenient and/or expensive for me; Amazon wants about $40 which is a bit more than my interest level. --RDBury (talk) 18:01, 12 August 2022 (UTC)[reply]

I don't know if I'm allowed to advocate for such things here per se, but I've checked Libgen, and you can indeed find it there. GalacticShoe (talk) 22:43, 12 August 2022 (UTC)[reply]

The second edition is available for free on Richard's website (or at least used to be). JBL (talk) 17:30, 13 August 2022 (UTC)[reply]

Seven years ago you had access to a copy: Wikipedia:Reference_desk/Archives/Mathematics/2015_May_22. --JBL (talk) 17:32, 13 August 2022 (UTC)[reply]

That may have been an on-line version mentioned above. I used to own a copy a couple of decades ago, but I have no idea where it is now. It's not the current edition anyway. --RDBury (talk) 18:49, 13 August 2022 (UTC)[reply]

Actually I did find an online version here. --RDBury (talk) 22:07, 13 August 2022 (UTC)[reply]