A355522 Triangle read by rows where T(n,k) is the number of reversed integer partitions of n with maximal difference k, if singletons have maximal difference 0.

2, 2, 1, 3, 1, 1, 2, 3, 1, 1, 4, 3, 2, 1, 1, 2, 6, 3, 2, 1, 1, 4, 6, 6, 2, 2, 1, 1, 3, 10, 6, 5, 2, 2, 1, 1, 4, 11, 11, 6, 4, 2, 2, 1, 1, 2, 16, 13, 10, 5, 4, 2, 2, 1, 1, 6, 17, 19, 12, 9, 4, 4, 2, 2, 1, 1, 2, 24, 24, 18, 11, 8, 4, 4, 2, 2, 1, 1

Offset

2,1

Comments

The triangle starts with n = 2, and k ranges from 0 to n - 2.

Links

Table of n, a(n) for n=2..79.

Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Example

Triangle begins:
  2
  2  1
  3  1  1
  2  3  1  1
  4  3  2  1  1
  2  6  3  2  1  1
  4  6  6  2  2  1  1
  3 10  6  5  2  2  1  1
  4 11 11  6  4  2  2  1  1
  2 16 13 10  5  4  2  2  1  1
  6 17 19 12  9  4  4  2  2  1  1
  2 24 24 18 11  8  4  4  2  2  1  1
  4 27 34 22 17 10  7  4  4  2  2  1  1
  4 35 39 33 20 15  9  7  4  4  2  2  1  1
  5 39 56 39 30 19 14  8  7  4  4  2  2  1  1
For example, row n = 8 counts the following reversed partitions:
  (8)         (233)      (35)      (125)    (26)    (116)  (17)
  (44)        (1223)     (134)     (11114)  (1115)
  (2222)      (11123)    (224)
  (11111111)  (11222)    (1124)
              (111122)   (1133)
              (1111112)  (111113)

Mathematica

Table[Length[Select[Reverse/@IntegerPartitions[n], If[Length[#]==1, 0, Max@@Differences[#]]==k&]], {n, 2, 15}, {k, 0, n-2}]

Crossrefs

Crossrefs found in the link are not repeated here.

Leading terms are A000005.

Row sums are A000041.

Counts m such that A056239(m) = n and A286470(m) = k.

This is a trimmed version of A238353, which extends to k = n.

For minimum instead of maximum we have A238354.

Ignoring singletons entirely gives A238710.

A001522 counts partitions with a fixed point (unproved), ranked by A352827.

A115720 and A115994 count partitions by their Durfee square.

A279945 counts partitions by number of distinct differences.

Cf. A064428, A091602, A179254, A238352, A239455, A286469, A325404, A355524, A355526, A355532.

Sequence in context: A239319 A236468 A116685 * A268190 A241150 A051135

Adjacent sequences: A355519 A355520 A355521 * A355523 A355524 A355525

Keyword

nonn,tabl

Author

Gus Wiseman, Jul 08 2022

Status

approved