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SmallerBeforeCloserPairs on SetPartition ​

Pairs a < b in different blocks with b the greatest element of its block.

On SetPartition

Called as CombinatorialStat(x, "SmallerBeforeCloserPairs"). The head's page has its examples.

Live

Distribution

Over SetPartitions at each size, tallied here in your browser; sizes past 10 are not offered.

FindStat St000579

The number of occurrences of the pattern {{1},{2}} such that 2 is a maximal element.

ObjectValue
{{1,2}}0
{{1,2,3}}0
{{1},{2}}1
{{1,2,3,4}}0
{{1,2},{3}}2
{{1,3},{2}}2
{{1},{2,3}}1
{{1,2,3,4,5}}0

Data from FindStat, the combinatorial statistic finder; the page at findstat.org/St000579 has the rest.