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OccurrencesOf1Over2Over3Openers13Closers2 on SetPartition ​

Occurrences of the pattern 1/2/3 with 1 and 3 the least elements of their blocks and 2 the greatest element of its block.

On SetPartition

Called as CombinatorialStat(x, "OccurrencesOf1Over2Over3Openers13Closers2"). The head's page has its examples.

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Distribution

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

FindStat St000588

The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are minimal, 2 is maximal.

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

Data from FindStat, the combinatorial statistic finder; the page at findstat.org/St000588 has the rest.