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

0, 2, 4, 12, 24, 60, 110, 238, 436, 860, 1516, 2848, 4874, 8666, 14664, 25120, 41342, 69178, 111596, 181890, 289170, 461086, 720944, 1131358, 1743016, 2689332, 4094090, 6228298, 9364440, 14072828, 20926402, 31080270, 45767490, 67255096, 98095260, 142805322

Offset

1,2

Comments

From Andrew Howroyd, Sep 16 2019: (Start)

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

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)

Links

Table of n, a(n) for n=1..36.

Example

a(4) = 12 = sum of {1,5},{2,4},{3,3},{4,2},{5,1} = 4 + 2 + 0 + 2 + 4.
From Andrew Howroyd, Sep 16 2019: (Start)
Case n = 4: The partitions of 4 in canonical order are:
  1) [1,1,1,1]
  2) [2,1,1]
  3) [2,2]
  4) [3,1]
  5) [4]
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.
Summing the distance between each element and its dual gives:
  a(4) = |1-5| + |2-4| + |3-3| + |4-2| + |5-1| = 12.
(End)

Crossrefs

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

Sequence in context: A045687 A233411 A057422 * A331392 A100538 A303794

Adjacent sequences: A036042 A036043 A036044 * A036046 A036047 A036048

Keyword

nonn,nice

Author

Olivier Gérard

Extensions

a(31)-a(36) from Andrew Howroyd, Sep 16 2019

Status

approved