Let $ G le S_n $ to be a finite permutation group with generators $ g_1, ldots, g_k $. We watch the action of $ G $ on the subsets

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

and $ 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 diagram. Together $ varphi (0, ldots, 0) = {1 } $ and

$$ 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} $$

It 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 on the label of its predecessor points, that is to say say these points which are one unit less in a single coordinate. For example if $ G = S_3 $ with $ g_1 = (1 2) $ and $ 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} $$

etc.

Now a row or column is stable at a certain value $ N $, if no new set appears along this row or column of $ N $ leave, for example the $ j $-th

line would be stable at $ N $ if $ { varphi (i, j): 0 le i le N } = { varphi (i, j): i ge 0 } $. As $ mathcal P ( {1, ldots, n }) $ is finished, each row or column will be stable from a given point in time.

But we could choose a "global" $ N $ so that each row or column will be stable after this point, or say differently if we search for each row or column $ N $ points right or up, each set that appears as labels among these rows or columns will appear among these first $ N $ Labels.

This could be seen by the following argument, if we choose any line for example, then the labels have the form that the singleton sets appear first, then sets of one greater cardinality, in the worst case of sets with two elements, then with three and soon. And if the size of the subsets does not increase, it traverses subsets of the same size. By this simple observation, we see that after at most $ binom {n} {1} + binom {n} {2} + ldots + binom {n} {n} = 2 ^ n $ steps, we saw each subset. The same reasoning applies to the columns,

and $ N = 2 ^ n $ would be such a "global" $ N $.

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

I hope my explanation is clear, let me know if something is unclear!