A381463 -id:A381463 - cp's OEIS frontend

A381462 Limiting sequence of the possible number of inversions in stable configurations of 3^n-1 chips in a chip firing-game directed 3-ary tree resulting from a permutation-based strategy of firing chips.

0, 1, 3, 4, 5, 9, 10, 12, 13, 14, 15, 16, 17, 18, 27, 28, 30, 31, 32, 36, 37, 39, 40, 41, 42, 43, 44, 45, 46, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 81, 82, 84, 85, 86, 90, 91, 93, 94, 95, 96, 97, 98, 99, 108, 109, 111, 112, 113, 117, 118, 120, 121, 122, 123, 124, 125

Offset: 1

Ryota Inagaki, Tanya Khovanova, and Austin Luo, Feb 24 2025

nonn

Consider a 3-ary, rooted infinite directed tree where each vertex has outdegree 3. Consider the chip firing game on this tree defined in Section 2 of Inagaki, Khovanova, and Luo (2025) with 3^n chips, which are labeled 0, 1, 2, ..., 3^n-1, at the root vertex.
Let A(3, n) be the increasing sequence of all possible numbers of inversions in stable configurations in a chip-firing game on a 3-ary tree starting with 3^n chips resulting from applying a permutation-based strategy corresponding to permutation w of 1,2,..., n. In the strategy, for each i = 1, 2, ..., n, chips with j as the w_i-th most significant digit are sent to the (j+1)-st leftmost child of the fired vertex. For each n, divide each element in A(3, n) by 9^n and put the resulting elements in order from smallest to greatest. These are the first several terms of the sequence.
This sequence is defined at the end of Section 4.3 of "Permutation-based Strategies for Labeled Chip-Firing on k-ary Trees."

Wikipedia, Chip-firing game
Ryota Inagaki, Tanya Khovanova, and Austin Luo, Permutation-based Strategies for Labeled Chip-Firing on k-ary Trees, arXiv:2503.09577 [math.CO], 2025.

Cf. A376116, A381463.

k = 3
s = set()
for i in range(2):
    for j in range(3):
        for l in range(4):
            for m in range(5):
                for n in range(6):
                    s.add(((k ** 5 - k ** (5-n)) + (k ** 4 - k ** (4-m)) + (k ** 3 - k ** (3-l)) + (k ** 2 - k ** (2-j))+ (k ** 1 - k ** (1-i)))// (k-1))
l = list(s)
l.sort()
print(l)

cp's OEIS Frontend

A381462 Limiting sequence of the possible number of inversions in stable configurations of 3^n-1 chips in a chip firing-game directed 3-ary tree resulting from a permutation-based strategy of firing chips.

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