A366173 Triangle of coefficients of Caylerian polynomials.

1, 1, 1, 2, 1, 8, 4, 1, 24, 42, 8, 1, 64, 276, 184, 16, 1, 162, 1458, 2298, 732, 32, 1, 400, 6844, 21232, 16000, 2752, 64, 1, 976, 29952, 164680, 240350, 99756, 9992, 128, 1, 2368, 125468, 1142952, 2882300, 2320008, 578420, 35488, 256

Offset: 0

Author

Michel Marcus, Oct 03 2023

Keywords

• nonn
• tabf

Examples

			Triangle begins:
  1
  1
  1 2
  1 8 4
  1 24 42 8
  1 64 276 184 16
  ...
Because polynomials are: 1; 1; 1 + 2t; 1 + 8t + 4t^2; 1 + 24t + 42t^2 + 8t^3; 1 + 64t + 276t^2 + 184t^3 + 16t^4; ...
		

Links

• Giulio Cerbai and Anders Claesson, Caylerian polynomials, arXiv:2310.01270 [math.CO], 2023. See p. 11.
• Giulio Cerbai and Anders Claesson, Enumerative aspects of Caylerian polynomials, arXiv:2411.08426 [math.CO], 2024. See p. 2.

Crossrefs

Cf. A000670 (row sums), A365449 (alternating row sums). Column 1 seems to be twice A048776.

Programs

Python

from itertools import product
def cayley_permutations(n):
    return [p for p in product(range(n), repeat=n) if len(set(p)) == max(p)+1]
for n in range(1, 9):
    a = [0] * n
    for p in cayley_permutations(n):
        a[sum(x>y for x,y in zip(p, p[1:]))] += 1
    print(a[::-1]) # Andrei Zabolotskii, Jul 26 2025

Extensions

Rows 6-9 from Andrei Zabolotskii, Jul 26 2025