A385574 Number of integer partitions of n with the same number of adjacent equal parts as adjacent unequal parts.

1, 1, 1, 1, 2, 3, 2, 4, 5, 6, 10, 11, 13, 17, 20, 30, 36, 44, 55, 70, 86, 98, 128, 156, 190, 235, 288, 351, 409, 499, 603, 722, 863, 1025, 1227, 1461, 1757, 2061, 2444, 2892, 3406, 3996, 4708, 5497, 6430, 7595, 8835, 10294, 12027, 13971, 16252, 18887, 21878

Offset

0,5

Comments

These are also integer partitions of n with the same number of distinct parts as maximal anti-runs of parts.

Links

Christian Sievers, Table of n, a(n) for n = 0..1000

Formula

For a partition p, let s(p) be its sum, e(p) the number of equal adjacent pairs, and d(p) the number of distinct adjacent pairs. Then Sum_{p partition} x^s(p) * t^(e(p)-d(p)) = (Product_{i>=1} (1 + x^i/(t*(1-t*x^i))) - 1) * t + 1, so a(n) is the coefficient of x^n*t^0 of this expression. - Christian Sievers, Jul 18 2025

Example

The partition (5,3,2,1,1,1,1) has 4 runs ((5),(3),(2),(1,1,1,1)) and 4 anti-runs ((5,3,2,1),(1),(1),(1)) so is counted under a(14).
The a(1) = 1 through a(10) = 10 reversed partitions (A = 10):
  (1)  (2)  (3)  (4)    (5)    (6)    (7)    (8)      (9)      (A)
                 (112)  (113)  (114)  (115)  (116)    (117)    (118)
                        (122)         (133)  (224)    (144)    (226)
                                      (223)  (233)    (225)    (244)
                                             (11123)  (11124)  (334)
                                                      (11223)  (11125)
                                                               (11134)
                                                               (11224)
                                                               (11233)
                                                               (12223)

Mathematica

Table[Length[Select[Reverse/@IntegerPartitions[n], Length[Union[#]]==Length[Split[#, #2!=#1&]]&]], {n, 0, 30}]

Prog

(PARI) lista(n)=Vec(polcoef((prod(i=1, n, 1+x^i/(t*(1-t*x^i))+O(x*x^n))-1)*t+1, 0, t)) \\ Christian Sievers, Jul 18 2025

Crossrefs

The RHS is counted by A116608, rank statistic A297155.

The LHS is counted by A133121, rank statistic A046660.

For related inequalities see A212165, A212168, A361204.

For subsets instead of partitions see A217615, A385572, A385575.

These partitions are ranked by A385576.

A000041 counts integer partitions, strict A000009.

A007690 counts partitions with no singletons, complement A183558.

A034296 counts flat or gapless partitions, ranks A066311 or A073491.

A034839 counts subsets by number maximal runs, for partitions A384881, strict A116674.

A047993 counts partitions with max part = length (A106529).

A098859 counts Wilf partitions (complement A336866), compositions A242882.

A268193 counts partitions by maximal anti-runs, strict A384905, subsets A384893.

A355394 counts partitions with neighbors, complement A356236.

Cf. A000071, A003114, A008284, A010027, A047966, A210034, A325324, A325325, A356606, A384882, A384885.

Sequence in context: A353853 A365384 A350840 * A176789 A308158 A325349

Adjacent sequences: A385571 A385572 A385573 * A385575 A385576 A385577

Keyword

nonn

Author

Gus Wiseman, Jul 04 2025

Status

approved