A381463 Limiting sequence of the possible number of inversions in stable configurations of 4^n-1 chips in a chip firing-game directed 4-ary tree resulting from a permutation-based strategy of firing chips.

0, 1, 4, 5, 6, 16, 17, 20, 21, 22, 24, 25, 26, 27, 64, 65, 68, 69, 70, 80, 81, 84, 85, 86, 88, 89, 90, 91, 96, 97, 100, 101, 102, 104, 105, 106, 107, 108, 109, 110, 111, 112, 256, 257, 260, 261, 262, 272, 273, 276, 277, 278, 280, 281, 282, 283, 320, 321, 324, 325, 326, 336, 337, 340

Offset: 1

Author

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

Keywords

• nonn

Comments

Consider a 4-ary, rooted infinite directed tree where each vertex has outdegree 4. A chip firing game on this tree is defined as in Section 2 of Inagaki, Khovanova, and Luo (2025). Here we start with 4^n chips labeled 0,1, ..., 4^n-1 at the root.

Let A(4, n) be the increasing sequence of all possible numbers of inversions in stable configurations in a chip-firing game on a directed regular 4-ary tree starting with 4^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 sent to the (j+1)-th leftmost child of the fired vertex. For each n divide each element in A(4, n) by 4^(n-1) * 9 and put the resulting elements in order from smallest to greatest. These are the first several terms of the sequence.

This sequence was defined at the end of Section 4.3 of Inagaki, Khovanova, and Luo (2025).

Links

• 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.

Crossrefs

Cf. A376116, A381462.

Programs

Python

k = 4
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)