Additive Macaulay posets (Q1368663)

For finite sequences \(a=(a_1,\dots, a_j)\) and \(b=(b_1,\dots, b_p)\) the juxtaposition of \(a\) and \(b\) is defined to be the sequence \((a_1,\dots, a_j,b_1,\dots, b_p)\). The sequences \(a\) and \(b\) are said to be overlapping if, for some \(s\leq\min\{j, p\}\), the inequalities \(a_{j-s+i}\geq b_i\) hold for all \(i=1,\dots, s\). A finite sequence of integers is called additive if the sum of any consecutive number of its terms is less than or equal to the sum of the same number of initial terms in the sequence and greater than or equal to the sum of the same number of final terms in the sequence. The following basic lemma is proved: The juxtaposition of overlapping additive sequences is additive. Two applications for chain products are given, a new proof (by induction) of the additivity of sequences which arise from shadows and a new result on the average rank of with respect to lexicographic order consecutive elements.