Confused between cyclic sum and symmetric sums.
four variables $a, b, c, d$ are given, what is the symmetric and cyclic sum?
I thought:
$$\sum_{cyc} ab = ab + ac + ad + bc + bd + cd$$
And
$$\sum_{sym} ab = 2(ab + ac + ad + bc + bd + cd)$$
Which one is correct?
$\endgroup$ 111 Answer
$\begingroup$Cyclic summations proceed systematically replacing the letters a, b, c, d in a cyclic manner as follows: $a->b->c->d->a$ therefore, defined over the ordered arrangement (a, b, c, d), $\sum_{cyc}{ab}=ab+bc+cd+da$.
Symmetric summation requires summing over all distinct arrangements as follows: $\sum_{symm}{ab}=ab+ac+ad+ba+bc+bd+ca+cb+cd+da+db+dc=2(ab+ac+ad+bc+bd+cd)$.
Defined over an ordered set of n distinct elements, the cyclic summation $\sum{ab}$ has n terms, whereas the symmetric summation has $n(n-1)$ terms (which may simplify into fewer terms).
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