Finding the reduced permutations of six numbers

I am trying to use Mathematica to find the following set of permutations:

$$\sum_{\rho\in \tilde{S}_{n+2}} {\rm sgn}(\rho)\eta^{\mu_{\rho_{(1)}}\mu_{\rho_{(2)}}}$$ where $$\tilde{S}_{n+2}:=\{\rho\in S_{n+2}: \rho_{(1)}<\rho_{(2)}\}$$ is the the reduced set of all permutations of $n+2$ numbers, and ${\rm sgn}$ is the sign function. Now I want to have it for 6 numbers, to use mathematica to find the set of all reduced permutations of {1,2,3,4,5,6}. Probably I should use DeleteCases, but I do not know how to put the condition in the reduced permutation. Any ideas?

• You're going to have to help us with the notation a bit. (1) You say "following set of permutations" when what I see is a sum over a set of permutations. (2) What is $\eta$? What is $\mu$? What is the notation $\rho_{(1))}$? (does it mean the permutation $\rho$ applied to element $1$)? What does $\mu_{\rho_{(1)}}$ mean? I actually want to help with this problem, because my gut reaction is that it could be interesting and fun to solve, but as it stands, I have no idea what you're asking for. – march Dec 7 '15 at 4:33
• Anyway. Maybe this: reducedPerms = Cases[Permutations@Range[6], a_ /; a[[1]] < a[[2]]]. And this: signs = Signature /@ reducedPerms. Or: reducedPerms = Select[Permutations@Range[6], #[[1]] < #[[2]]&.] – march Dec 7 '15 at 4:58
• Sorry, you are right it is a sum over all permutations...$\eta$ is simply the Kronecker delta, $\mu$ is an index for the member of the set...and $\rho{(1)}$ is the first member of each permutation set, let say for example $\{1,2,3,4,5,6\}$ then $\rho_{(1)}=1$ etc – Naser Dec 7 '15 at 9:22
• we can simply forget about the first equation, the second one is related to my question, if you have a sum over a set of permutations and you have to exclude some of them, in this case I just need those which the first member is smaller than the second one, and the other four could be anything.. – Naser Dec 7 '15 at 9:31
• What you just suggested seems to work march, thanks – Naser Dec 7 '15 at 9:40

We start by generating all permutations and selecting those that satisfy the condition. There are two possibilities, one using Select and one using Cases. Take your pick.

n = 6;
reducedPerms = Cases[Permutations@Range[n], a_ /; a[[1]] < a[[2]]];
reducedPerms = Select[Permutations@Range[n], #[[1]] < #[[2]] &];


We can get the Signature of a permutation by using Signature:

Signature[{1,2,3,5,4,6}]
(* -1 *)

• Another possibility: Select[Permutations[Range[6]], OrderedQ[Take[#, 2]] &]. – J. M.'s discontentment Dec 9 '15 at 5:45