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In the theory of cumulants of vector-valued random variables, the following types of formulas appear: \begin{equation} \theta^i \theta^{jk} [3] = \theta^i \theta^{jk} + \theta^j \theta^{ik} + \theta^k \theta^{ij} \end{equation} This is supposed to mean as follows: sum over all possible permutations (in this case 3) of the indices. The indices of a given theta are symmetric and need not be counted twice.

Another example would be $\theta^i \theta^j \theta^{kl}$ [6]. In this case there is a total of 6 terms:

$i\;|\;j\;|\;kl$

$i\;|\;k\;|\;jl$

$i\;|\;l\;|\;jk$

$j\;|\;k\;|\;il$

$j\;|\;l\;|\;ik$

$k\;|\;l\;|\;ij$

Going to even higher order, $\theta^{ij} \theta^{kl} \theta^{mn}$ [15] has 15 terms, and etc.

I am unaware of any closed expressions for the number of terms. Thus, I was trying to use Mathematica to numerically compute all terms involved in a given permutation.

However, so far I have been entirely unsuccessful. Any ideas?

Here are some (failed) attempts, made with 4 indices and the combination $\theta^i \theta^j \theta^{kl}$. This lists all permutations of four terms. 

perm = Permutations[{a, b, c, d}]

I can use Mr. Wizard's dynP to partition in the size I want. For instance, 

dynP[{a,b,c,d},{1,1,2}]
{{a},{b},{c,d}}

So, an alternative, would be to do this for all members of perm, and then eliminate duplicates according to the above-specified criteria. This, however, I failed to do.

I appreciate any help in advance.

Best regards,

Gabriel

J. M.'s missing motivation
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Gabriel Landi
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3 Answers3

12

If I'm not mistaken, we can do this in closed form. Suppose we have $n$ indices split up into $n_1$ one-index variables, $n_2$ two-index variables, etc. Then the total number of distinct permutations is just $N=\dfrac{n!}{\prod\limits_{i}n_i!(i!)^{n_i}}$.

This accounts for the $n_i!$ permutations of $i$-index variables and the $i!$ permutations of each of their indices.

This can be implemented pretty simply. With x being a list of the $n_i$ (including zeros),

numDistinct[x_] :=
  Plus@@(x.Range@Length@x)!/Product[x[[i]]! (i!)^x[[i]], {i, Length@x}]

This yields numDistinct[{2, 1}] = 6 and numDistinct[{0, 3}] = 15, as it should.

m_goldberg
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Eric Thewalt
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5

Sorry for this seat-of-the-pants approach. I was too rushed to figure out the underlying combinatorics. f produces all the permutations (including equivalent cases), sorts them at Level 2, and then delete duplicates. dynP is by Mr.Wizard.

dynP[l_,p_]:=MapThread[l[[#;;#2]]&,{{0}~Join~Most@#+1,#}&@Accumulate@p]

f[p_]:=DeleteDuplicates[Map[Sort(dynP[#,p]&/@
  Permutations[FromCharacterCode/@Range[97,96+Plus@@p]]),2]]//Length


f /@ {{1, 1, 3}, {2, 2, 2}, {1, 2, 3}, {3, 1, 2}}

{10, 15, 60, 60}

Remove //Length from function to see the permutations.

Community
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DavidC
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  • This is precisely the type of Sort syntax that I was trying to figure out. Thanks for the help. – Gabriel Landi Mar 13 '13 at 02:30
  • Glad to hear that. Sort seemed to be the key to avoiding equivalent permutations (equivalent in a special sense here). There are likely to be more elegant or efficient ways of doing it. – DavidC Mar 13 '13 at 02:37
4
ClearAll[numtermsF];
numtermsF = Coefficient[MomentConvert[Moment[Total@{##}], Cumulant], 
  Times @@ (Cumulant[#[[1]]]^(#[[2]]) & /@ Tally[{##}])] &
numtermsF @@ # & /@ {{1, 1, 2}, {2, 2, 2}, {1, 1, 3}, {3, 1, 2}, {1, 2, 3}}
(* {6, 15, 10, 60, 60} *)

see also: Documentation Center MomentConvert>Applications>Combinatorial Uses of MomentConvert

kglr
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