SecondAlladiWeight on IntegerPartition

Half the last part (rounded up if odd), times the product over consecutive parts of half the gap less 1 for each even one of the two.

On IntegerPartition

Alladi's second weight: for each part, delta is 1 if it is even and 0 if odd; the factor for the last part is (last + 1 - delta)/2 and for each consecutive pair (gap - delta - delta')/2.

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

Live

Distribution

Over IntegerPartitions at each size, tallied here in your browser; sizes past 30 are not offered.

Sizes 1 to

FindStat St000478

Another weight of a partition according to Alladi.

Object	Value
[2]	1
[3]	2
[4]	2
[5]	3
[6]	3
[7]	4
[8]	4
[9]	5

Data from FindStat, the combinatorial statistic finder; the page at findstat.org/St000478 has the rest.