A minimization problem concerning subsets of a finite set

Characterizing Profiles ofk-Families in Additive Macaulay Posets

“…It is now easy to see that 1, 0, 2, 1, 0 is the profile of a 2-family in A 4 , the corresponding canonical antichain in 2A 4 being [113,132,142,200] and the corresponding canonical 2-family in A 4 being [13,32,42,00]. Similarly, 1, 1, 2, 1, 0 is not the profile of a 2-family since``there is insufficient room'' in 2A 4 for the corresponding canonical antichain.…”

“…We illustrate in the context of checking 2-families in A 4 . We abbreviate the elements (c, (i, j )) of 2A 4 to cij.…”

“…We have written the elements of 2A 4 by rank with Macaulay order increasing from top to bottom and we have staggered the ranks so that if x is the last element of an initial segment of rank j, then the shadow of that segment is the initial segment of rank j&1 consisting of all elements not lower than x (Fig. 1).…”

“…This property of the function |2F( }, P l )| is referred to as subadditivity [4]. Engel [16] develops a property equivalent to additivity which he calls little-submodularity.…”

“…Clements [4] and independently Daykin et al [15] characterized the profiles of antichains in S(k 1 , k 2 , ..., k n ) and S(1, 1, ..., 1) respectively. Then Daykin characterized the profiles of k-families in S(1, 1, ..., 1) [14] and Clements [6,7] verified that his results go over to S(k 1 , k 2 , ..., k n ).…”