A395081 Triangle read by rows: row n (n >= 1) gives the concatenated run-length encoding of the integer partitions of n, sorted as in A334442.

1, 1, 2, 1, 1, 2, 3, 1, 1, 1, 2, 1, 1, 3, 4, 1, 1, 1, 3, 1, 2, 2, 1, 2, 2, 1, 1, 4, 5, 1, 1, 1, 4, 1, 2, 1, 3, 1, 1, 2, 3, 1, 1, 1, 2, 2, 1, 3, 2, 1, 1, 5, 6, 1, 1, 1, 5, 1, 2, 1, 4, 1, 3, 2, 1, 2, 4, 1, 1, 1, 2, 1, 3, 1, 2, 3, 1, 3, 3, 1, 1, 2, 2, 2, 1, 4, 2, 1, 1, 6

Offset

1,3

Comments

Each partition lambda is represented as a flat list [v_1, m_1, v_2, m_2, ...] where v_i is the part value and m_i is its multiplicity.

Links

Table of n, a(n) for n=1..90.

Example

  [1] [1, 1]
  [2] [2, 1, 1, 2]
  [3] [3, 1, 1, 1, 2, 1, 1, 3]
  [4] [4, 1, 1, 1, 3, 1, 2, 2, 1, 2, 2, 1, 1, 4]
  [5] [5, 1, 1, 1, 4, 1, 2, 1, 3, 1, 1, 2, 3, 1, 1, 1, 2, 2, 1, 3, 2, 1, 1, 5]

Prog

(Python)
def A395081(n: int) -> list[list[int]]:
    """ Returns a list of lists, where each inner list is a flat representation:
    [value_1, multiplicity_1, value_2, multiplicity_2, ...] """
    def gen(n, i):  # Generator
        if n == 0: yield []
        elif i == 1: yield [1, n]
        else:
            if n >= i:
                for p in gen(n - i, min(n - i, i)):
                    if p and p[-2] == i:
                        new_p = list(p)
                        new_p[-1] += 1
                        yield new_p
                    else:
                        yield p + [i, 1]
            yield from gen(n, i - 1)
    if n == 0: return [[]]
    partitions = list(gen(n, n))
    # Sort by length: length is sum of multiplicities (every second element)
    partitions.sort(key=lambda p: sum(p[1::2]))
    return partitions
A395081row = lambda n: [p for parts in A395081(n) for p in parts] # flatten
for n in range(1, 8): print([n], A395081row(n))

Crossrefs

Cf. A334442, A395698.

Sequence in context: A309736 A368010 A237453 * A383134 A265754 A089309

Adjacent sequences: A395078 A395079 A395080 * A395082 A395083 A395084

Keyword

nonn,tabf

Author

Peter Luschny, May 02 2026

Extensions

Name edited and offset set to 1 by Peter Luschny, May 13 2026

Status

approved