A351203 Number of integer partitions of n of whose permutations do not all have distinct runs.

0, 0, 0, 0, 1, 2, 3, 6, 11, 16, 24, 36, 52, 73, 101, 135, 184, 244, 321, 418, 543, 694, 889, 1127, 1427, 1789, 2242, 2787, 3463, 4276, 5271, 6465, 7921, 9655, 11756, 14254, 17262, 20830, 25102, 30152, 36172, 43270, 51691, 61594, 73300, 87023, 103189, 122099, 144296, 170193, 200497

Offset

0,6

Links

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Mathematics Stack Exchange, What is a sequence run? (answered 2011-12-01)

Formula

a(n) = A000041(n) - A351204(n). - Andrew Howroyd, Jan 27 2024

Example

The a(4) = 1 through a(9) = 16 partitions:
  (211)  (221)  (411)    (322)    (332)      (441)
         (311)  (2211)   (331)    (422)      (522)
                (21111)  (511)    (611)      (711)
                         (3211)   (3221)     (3321)
                         (22111)  (3311)     (4221)
                         (31111)  (4211)     (4311)
                                  (22211)    (5211)
                                  (32111)    (22221)
                                  (41111)    (32211)
                                  (221111)   (33111)
                                  (2111111)  (42111)
                                             (51111)
                                             (222111)
                                             (321111)
                                             (2211111)
                                             (3111111)
For example, the partition x = (2,1,1,1,1) has the permutation (1,1,2,1,1), with runs (1,1), (2), (1,1), which are not all distinct, so x is counted under a(6).

Mathematica

Table[Length[Select[IntegerPartitions[n], MemberQ[Permutations[#], _?(!UnsameQ@@Split[#]&)]&]], {n, 0, 15}]

Prog

(Python)
from sympy.utilities.iterables import partitions
from itertools import permutations, groupby
from collections import Counter
def A351203(n):
    c = 0
    for s, p in partitions(n, size=True):
        for q in permutations(Counter(p).elements(), s):
            if max(Counter(tuple(g) for k, g in groupby(q)).values(), default=0) > 1:
                c += 1
                break
    return c # Chai Wah Wu, Oct 16 2023

Crossrefs

The version for run-lengths instead of runs is A144300.

The version for normal multisets is A283353.

The Heinz numbers of these partitions are A351201.

The complement is counted by A351204.

A005811 counts runs in binary expansion.

A044813 lists numbers whose binary expansion has distinct run-lengths.

A059966 counts Lyndon compositions, necklaces A008965, aperiodic A000740.

A098859 counts partitions with distinct multiplicities, ordered A242882.

A297770 counts distinct runs in binary expansion.

A003242 counts anti-run compositions, ranked by A333489.

Counting words with all distinct runs:

- A351013 = compositions, for run-lengths A329739, ranked by A351290.

- A351016 = binary words, for run-lengths A351017.

- A351018 = binary expansions, for run-lengths A032020, ranked by A175413.

- A351200 = patterns, for run-lengths A351292.

- A351202 = permutations of prime factors.

Cf. A000041, A035363, A047993, A116608, A238130 or A238279, A325545, A329746, A350842, A351003, A351004, A351291.

Sequence in context: A298702 A076307 A102990 * A138519 A138520 A210458

Adjacent sequences: A351200 A351201 A351202 * A351204 A351205 A351206

Keyword

nonn

Author

Gus Wiseman, Feb 12 2022

Extensions

a(26) onwards from Andrew Howroyd, Jan 27 2024

Status

approved