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A356116 Triangle read by row. The reduced triangle of the partition_triangle A355776. 4
0, 0, 0, 0, 1, 0, 0, 5, 5, 0, 0, 16, 46, 16, 0, 0, 42, 252, 252, 42, 0, 0, 99, 1086, 2241, 1086, 99, 0, 0, 219, 4097, 15129, 15129, 4097, 219, 0, 0, 466, 14272, 87058, 154426, 87058, 14272, 466, 0, 0, 968, 47300, 452672, 1305062, 1305062, 452672, 47300, 968, 0 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
1,8
COMMENTS
By a partition triangle, we understand an irregular triangle where each row corresponds to a mapping of Partitions(n) -> ZZ. We assume a fixed order of the partitions given. Here we will use the ordering defined in A080577. Examples are A355776, A355777, and A134264. 'Reducing' then means summing the values corresponding to the partitions of n with length k. The 'reduced partition triangle' then is a regular triangle with T(n, k) with 1 <= k <= n.
Conversely, A355776, the statistic of permutations whose Lehmer code is nonmonotonic, can be seen as a refinement of this triangle, which in turn is a refinement of the sequence A056986, the number of permutations on [n] containing any given pattern alpha in the symmetric group S_3.
LINKS
Peter Luschny, Permutations with Lehmer, a SageMath Jupyter Notebook.
EXAMPLE
Triangle T(n, k) starts:
[1] [0]
[2] [0, 0]
[3] [0, 1, 0]
[4] [0, 5, 5, 0]
[5] [0, 16, 46, 16, 0]
[6] [0, 42, 252, 252, 42, 0]
[7] [0, 99, 1086, 2241, 1086, 99, 0]
[8] [0, 219, 4097, 15129, 15129, 4097, 219, 0]
[9] [0, 466, 14272, 87058, 154426, 87058, 14272, 466, 0]
[10][0, 968, 47300, 452672, 1305062, 1305062, 452672, 47300, 968, 0]
.
Row 6 of the partition triangle A355776 is:
[0, [10, 20, 12], [61, 162, 29], [102, 150], 42, 0]
Adding the bracketed terms reduces this row to row 6 of the above triangle.
PROG
(SageMath)
from functools import cache
@cache
def Pn(n: int, k: int) -> int:
if k == 0: return 0
if n == 0 or k == 1: return 1
return Pn(n, k - 1) + Pn(n - k, k) if k <= n else Pn(n, k - 1)
def reduce_parts(fun, n: int) -> list[int]:
funn: list[int] = fun(n)
return [sum(funn[Pn(n, k):Pn(n, k + 1)]) for k in range(n)]
def reduce_partition_triangle(fun, n: int) -> list[list[int]]:
return [reduce_parts(fun, k) for k in range(1, n)]
reduce_partition_triangle(A355776_row, 6)
CROSSREFS
A002662 (column 1), A056986 (row sums), A355776 (refinement).
Sequence in context: A200506 A285070 A285288 * A281165 A282481 A286979
KEYWORD
nonn,tabl
AUTHOR
Peter Luschny, Jul 28 2022
STATUS
approved

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Last modified March 29 11:45 EDT 2024. Contains 371278 sequences. (Running on oeis4.)