A356265 -id:A356265 - cp's OEIS frontend

A356264 Partition triangle read by rows, counting reducible permutations, refining triangle A356265.

0, 0, 1, 0, 1, 2, 0, 1, 5, 3, 2, 0, 1, 9, 12, 15, 10, 2, 0, 1, 14, 23, 12, 47, 94, 11, 31, 24, 2, 0, 1, 20, 38, 48, 113, 293, 154, 137, 183, 409, 78, 63, 54, 2, 0, 1, 27, 60, 87, 49, 227, 738, 883, 451, 457, 670, 2157, 1007, 1580, 79, 605, 1520, 384, 127, 116, 2, 0

Offset: 0

Peter Luschny, Aug 05 2022

			[0] 0;
[1] 0;
[2] 1, 0;
[3] 1, 2, 0;
[4] 1, [5, 3], 2, 0;
[5] 1, [9, 12], [15,  10],   2,  0;
[6] 1, [14, 23, 12], [ 47,  94, 11], [31,   24],   2,  0;
[7] 1, [20, 38, 48], [113, 293, 154, 137], [183, 409, 78], [63, 54], 2, 0;
Summing the bracketed terms reduces the triangle to A356265.

Peter Luschny, Permutations with Lehmer, a SageMath Jupyter Notebook.

Cf. A356265 (reduced), A356262, A356263, A356291 (row sums).

import collections
def reducible(p) -> bool:  # p is a Sage-Permutation
    return any(i for i in range(1, p.size())
                  if all(p(j) < p(k)
                      for j in range(1, i + 1)
                          for k in range(i + 1, p.size() + 1) ) )
def void(L) -> bool: return True
def perm_red_stats(n: int, part_costraint, lehmer_constraint):
    res = collections.defaultdict(int)
    for p in Permutations(n):
        if not part_costraint(p): continue
        l: list[int] = p.to_lehmer_code()
        if lehmer_constraint(l):
            c: list[int] = [l.count(i) for i in range(len(p)) if i in l]
            res[Partition(reversed(sorted(c)))] += 1
    return sorted(res.items(), key=lambda x: len(x[0]))
@cache
def A356264_row(n: int) -> list[int]:
    if n < 2: return [0]
    return [v[1] for v in perm_red_stats(n, reducible, void)] + [0]
def A356264(n: int, k: int) -> int:
    return A356264_row(n)[k]
for n in range(0, 8): print(A356264_row(n))

A357079 Triangle read by rows. T(n, k) = A356265(n, k) + A357078(n, k) for 0 <= k <= n.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 3, 2, 1, 0, 9, 12, 2, 1, 0, 49, 37, 31, 2, 1, 0, 329, 149, 176, 63, 2, 1, 0, 2561, 794, 853, 702, 127, 2, 1, 0, 22369, 5599, 3836, 5709, 2549, 255, 2, 1, 0, 216225, 47275, 18422, 37609, 33949, 8886, 511, 2, 1, 2291457, 451176, 107535, 218506, 344670, 184653, 29777, 1023, 2, 1

Offset: 0

Peter Luschny, Sep 11 2022

The triangle is the termwise sum of a refinement of the number of irreducible permutations A357078, and A356265, which is a refinement of the number of reducible permutations.

			Triangle T(n, k) starts:                                 [Row sums]
[0] 1;                                                       [1]
[1] 0,     1;                                                [1]
[2] 0,     1,      1;                                        [2]
[3] 0,     3,      2,     1;                                 [6]
[4] 0,     9,     12,     2,     1;                          [24]
[5] 0,    49,     37,    31,     2,     1;                   [120]
[6] 0,   329,    149,   176,    63,     2,    1;             [720]
[7] 0,   2561,   794,   853,   702,   127,    2,   1;        [5040]
[8] 0,  22369,  5599,  3836,  5709,  2549,  255,   2, 1;     [40320]
[9] 0, 216225, 47275, 18422, 37609, 33949, 8886, 511, 2, 1;  [362880]

Cf. A356265, A357078, A059438, A003319.

def A357079_triangle(dim):
    T = A357078_triangle(dim)
    return [[0^n] + [T[n][k+1] + A356265_row(n)[k]
           for k in range(n)] for n in range(dim)]
A357079_triangle(9)

A356115 Triangle read by rows. The reduced triangle of the partition triangle of reducible permutations with weakly decreasing Lehmer code (A356266). T(n, k) for n >= 1 and 0 <= k < n.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 3, 1, 1, 0, 4, 6, 3, 1, 0, 9, 20, 6, 6, 1, 0, 11, 45, 50, 15, 10, 1, 0, 19, 93, 185, 80, 36, 15, 1, 0, 22, 196, 462, 490, 161, 77, 21, 1, 0, 33, 312, 1120, 1834, 1050, 336, 148, 28, 1

Offset: 1

Peter Luschny, Aug 16 2022

			[ 1] [1]
[ 2] [0,  1]
[ 3] [0,  1,   1]
[ 4] [0,  3,   1,    1]
[ 5] [0,  4,   6,    3,    1]
[ 6] [0,  9,  20,    6,    6,    1]
[ 7] [0, 11,  45,   50,   15,   10,   1]
[ 8] [0, 19,  93,  185,   80,   36,  15,   1]
[ 9] [0, 22, 196,  462,  490,  161,  77,  21,  1]
[10] [0, 33, 312, 1120, 1834, 1050, 336, 148, 28, 1]

Peter Luschny, Permutations with Lehmer, a SageMath Jupyter Notebook.

Cf. A356266 (partition version), A356265, A120588 (row sums).

# uses function reduce_partition_triangle from A356265.
def A356115_row(n: int) -> list[int]:
    return reduce_partition_triangle(A356266_row, n + 1)[n - 1]
def A356115(n: int, k: int) -> int:
    return A356115_row(n)[k]
for n in range(1, 11):
    print([n], A356115_row(n))

A357078 Triangle read by rows. The partition transform of A355488, which are the alternating row sums of the number of permutations of [n] with k components (A059438).

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 2, 0, 1, 0, 8, 4, 0, 1, 0, 48, 16, 6, 0, 1, 0, 328, 100, 24, 8, 0, 1, 0, 2560, 688, 156, 32, 10, 0, 1, 0, 22368, 5376, 1080, 216, 40, 12, 0, 1, 0, 216224, 46816, 8456, 1504, 280, 48, 14, 0, 1, 0, 2291456, 450240, 73440, 11808, 1960, 348, 56, 16, 0, 1

Offset: 0

Peter Luschny, Sep 10 2022

The partition transform (also called De Moivre polynomials by Cormac O'Sullivan) is defined in the program section as a Sage script.

The triangle represents a refinement of the number of irreducible permutations, A003319. Together with the refinement of the number of reducible permutations A356265 the triangle sums to the refinement of the factorial numbers given in A357079.

			Triangle T(n, k) starts:                         [Row sums]
[0] 1;                                               [1]
[1] 0,      1;                                       [1]
[2] 0,      0,     1;                                [1]
[3] 0,      2,     0,    1;                          [3]
[4] 0,      8,     4,    0,    1;                    [13]
[5] 0,     48,    16,    6,    0,   1;               [71]
[6] 0,    328,   100,   24,    8,   0,  1;           [461]
[7] 0,   2560,   688,  156,   32,  10,  0,  1;       [3447]
[8] 0,  22368,  5376, 1080,  216,  40, 12,  0, 1;    [29093]
[9] 0, 216224, 46816, 8456, 1504, 280, 48, 14, 0, 1; [273343]

Peter Luschny, The P-transform.

Cf. A355488, A003319, A059438, A356265, A357079, A269941.

from functools import cache
def PartTrans(dim, f):
    X = var(['x' + str(i) for i in range(dim + 1)])
    @cache
    def PCoeffs(n: int, k: int):
        R = PolynomialRing(ZZ, X[1: n - k + 2], n - k + 1, order='lex')
        if k == 0: return R(k^n)
        return R(sum(PCoeffs(n - j, k - 1) * f(j)
                     for j in range(1, n - k + 2)))
    return [[PCoeffs(n, k) for k in range(n + 1)] for n in range(dim)]
def A357078_triangle(dim):
    A = ZZ[['t']]; g = A([0] + [factorial(n) for n in range(1, 30)]).O(dim+2)
    return PartTrans(dim, lambda n: list(g / (1 + 2 * g))[n])
A357078_triangle(9)

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A356264 Partition triangle read by rows, counting reducible permutations, refining triangle A356265.

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A357079 Triangle read by rows. T(n, k) = A356265(n, k) + A357078(n, k) for 0 <= k <= n.

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A356115 Triangle read by rows. The reduced triangle of the partition triangle of reducible permutations with weakly decreasing Lehmer code (A356266). T(n, k) for n >= 1 and 0 <= k < n.

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A357078 Triangle read by rows. The partition transform of A355488, which are the alternating row sums of the number of permutations of [n] with k components (A059438).

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