A383111 Number of integer partitions of n having more than one permutation with all distinct run-lengths.

0, 0, 0, 0, 1, 3, 3, 8, 9, 13, 17, 26, 27, 43, 51, 61, 78, 103, 115, 153, 174, 213, 255, 316, 354, 442, 508, 610, 701, 848, 950, 1153, 1303, 1539, 1750, 2075, 2318, 2738, 3081

Offset

0,6

Links

Table of n, a(n) for n=0..38.

Example

The partition (2,1,1) has two permutations with all distinct run-lengths: (1,1,2), (2,1,1), so it is counted under a(4).
The a(4) = 1 through a(9) = 13 partitions:
  (211)  (221)   (411)    (322)     (332)      (441)
         (311)   (3111)   (331)     (422)      (522)
         (2111)  (21111)  (511)     (611)      (711)
                          (2221)    (5111)     (3222)
                          (4111)    (22211)    (6111)
                          (22111)   (41111)    (22221)
                          (31111)   (221111)   (33111)
                          (211111)  (311111)   (51111)
                                    (2111111)  (222111)
                                               (411111)
                                               (2211111)
                                               (3111111)
                                               (21111111)

Mathematica

Table[Length[Select[IntegerPartitions[n], Length[Select[Permutations[#], UnsameQ@@Length/@Split[#]&]]>1&]], {n, 0, 15}]

Prog

(Python)
from itertools import groupby
from sympy.utilities.iterables import partitions, multiset_permutations
def a(n):
    c = 0
    for p in partitions(n):
        cp = 0
        for mp in multiset_permutations(p):
            passes, run_lengths = True, set()
            for k, g in groupby(mp):
                rl = len(list(g))
                if rl in run_lengths:
                    passes = False
                    break
                run_lengths.add(rl)
            if passes:
                cp += 1
            if cp > 1:
                c += 1
                break
    return c
print([a(n) for n in range(21)]) # Michael S. Branicky, Nov 02 2025

Crossrefs

For a unique choice we have A000005, ranks A000961.

For at least one choice we have A239455, ranks A351294, conjugate A381432.

For no choices we have A351293, ranks A351295, conjugate A381433.

The complement is A351293 + A000005, ranks too dense.

For equal instead of distinct run-lengths we have A383090, ranks A383089.

These partitions are ranked by A383113 = positions of terms > 1 in A382771.

Cf. A047966, A098859, A130091, A353837, A100471, A100881, A100882, A100883

A000041 counts integer partitions, strict A000009.

A008284 counts partitions by length, strict A008289.

A329738 counts compositions with equal run-lengths, ranks A353744.

Cf. A047993, A329739, A381541, A381636, A381717, A382857, A382915, A383013, A383092, A383094, A383097.

Sequence in context: A368738 A135477 A182473 * A363125 A335602 A092549

Adjacent sequences: A383108 A383109 A383110 * A383112 A383113 A383114

Keyword

nonn,hard,more

Author

Gus Wiseman, Apr 20 2025

Extensions

a(21)-a(38) from Jakub Buczak, May 04 2025

Status

approved