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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).
2
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
0,8
COMMENTS
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.
EXAMPLE
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]
PROG
(SageMath)
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)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Peter Luschny, Sep 10 2022
STATUS
approved