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 A036045 Sum of distances between dual pairs of partitions of n for the canonical order. 11

%I

%S 0,2,4,12,24,60,110,238,436,860,1516,2848,4874,8666,14664,25120,41342,

%T 69178,111596,181890,289170,461086,720944,1131358,1743016,2689332,

%U 4094090,6228298,9364440,14072828,20926402,31080270,45767490,67255096,98095260,142805322

%N Sum of distances between dual pairs of partitions of n for the canonical order.

%C From _Andrew Howroyd_, Sep 16 2019: (Start)

%C Canonical order means each partition is sorted in descending order of part size and then the partitions are listed in lexicographic order.

%C a(n) is even since for parts that are not self-dual the distance between a part and its dual is counted twice, and for parts that are self-dual the distance is zero. (End)

%e a(4) = 12 = sum of {1,5},{2,4},{3,3},{4,2},{5,1} = 4 + 2 + 0 + 2 + 4.

%e From _Andrew Howroyd_, Sep 16 2019: (Start)

%e Case n = 4: The partitions of 4 in canonical order are:

%e 1) [1,1,1,1]

%e 2) [2,1,1]

%e 3) [2,2]

%e 4) [3,1]

%e 5) [4]

%e Partitions [1,1,1,1] and [4] are dual, partitions [2,1,1] and [3,1] are dual and partition [2,2] is self-dual.

%e Summing the distance between each element and its dual gives:

%e a(4) = |1-5| + |2-4| + |3-3| + |4-2| + |5-1| = 12.

%e (End)

%Y Cf. A036046, A036047, A036048, A036049, A036050, A036051, A036052, A036053, A036054, A036055, A036056.

%K nonn,nice

%O 1,2