Leave $ G le S_n $ be a finite permutation group with generators $ g_1, ldots, g_k $. We look at the action of $ G $ in subsets

with $ A ^ g = { alpha ^ g: alpha in A } $ for $ A subseteq Omega $

Y $ g in G $. Label the grid $ mathbb N ^ k $ with subsets $ varphi: mathbb N ^ k to mathcal P ( {1, ldots, n }) $ according to the following scheme. Set $ varphi (0, ldots, 0) = {1 } $ Y

$$ begin {align}

varphi (n_1, ldots, n_k) & = varphi ( min (n_1-1,0), n_2, ldots, n_k) ^ {g_1}

\ & quad cup varphi (n_1, min (n_2-1,0), ldots, n_k) ^ {g_2}

\ & qquad qquad vdots

\ & quad cup varphi (n_1, n_2, ldots, min (n_ {k-1} -1,0), n_k) ^ {g_ {k-1}}

\ & quad cup varphi (n_1, n_2, ldots, n_ {k-1}, min (n_k-1,0)) ^ {g_k}.

end {align} $$

This means that we start with $ {1 } $ at the origin, and the label of any other point is the union of the action of the generators in the label of their predecessor points, that is, those points that are one unit less in a single coordinate. For example, yes $ G = S_3 $ with $ g_1 = (1 2) $ Y $ g_2 = (1 2 3) $

then $ varphi (0,0) = {1 } $

$$ begin {align}

varphi (1,0) & = {2 } \

varphi (2,0) & = {1 } \

varphi (0,1) & = {2 } \

varphi (0.2) & = {3 } \

varphi (0,3) & = {1 } \

varphi (1,1) & = varphi (0,1) cup varphi (1,0) = {2 } \

varphi (1,2) & = {2,3 }

end {align} $$

and so.

Now a row or column is stable at some value $ N $, if new sets do not appear along this row or column of $ N $ onwards, for example the $ j $-th

the row would be stable in $ N $ Yes $ { varphi (i, j): 0 le i le N } = { varphi (i, j): i ge 0 } $. How $ mathcal P ( {1, ldots, n }) $ It is finite, each row or column will be stable from some point forward.

But we could choose a "global" $ N $ so that each row or column is stable after this point, or put another way if we look for each row or column $ N $ point to the right or up, each set that appears as labels between these rows or columns will appear between these first $ N $ labels

This could be seen in the following argument, if we choose any row, for example, then the tags have the form that first singleton sets appear, then sets of some larger cardinality, in the worst case with two elements, then with three and soon. And if the size of the subsets does not increase, then it scrolls through some subsets of the same size. By this simple observation we see that after a lot $ binom {n} {1} + binom {n} {2} + ldots + binom {n} {n} = 2 ^ n $ Steps we have seen each subset. The same reasoning applies to the columns,

Y $ N = 2 ^ n $ it would be so "global" $ N $.

But somehow I have the feeling that it could be done much better than $ 2 ^ n $, incorporating the structure of the cycle and the structure of the cycle that we have for action in the subsets. But somehow I can't find a better formula. So, could we find a better upper limit for the "global" $ N $? Or at least some asymptotic?

I hope my explanation is clear, let me know if something is not clear!