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.

%I A381462 #55 Jul 23 2025 16:07:19
%S A381462 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,
%T A381462 43,44,45,46,48,49,50,51,52,53,54,55,56,57,58,81,82,84,85,86,90,91,93,
%U A381462 94,95,96,97,98,99,108,109,111,112,113,117,118,120,121,122,123,124,125
%N 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.
%C A381462 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.
%C A381462 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.
%C A381462 This sequence is defined at the end of Section 4.3 of "Permutation-based Strategies for Labeled Chip-Firing on k-ary Trees."
%H A381462 Wikipedia, <a href="https://en.wikipedia.org/wiki/Chip-firing_game">Chip-firing game</a>
%H A381462 Ryota Inagaki, Tanya Khovanova, and Austin Luo, <a href="https://arxiv.org/abs/2503.09577">Permutation-based Strategies for Labeled Chip-Firing on k-ary Trees</a>, arXiv:2503.09577 [math.CO], 2025.
%o A381462 (Python)
%o A381462 k = 3
%o A381462 s = set()
%o A381462 for i in range(2):
%o A381462     for j in range(3):
%o A381462         for l in range(4):
%o A381462             for m in range(5):
%o A381462                 for n in range(6):
%o A381462                     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))
%o A381462 l = list(s)
%o A381462 l.sort()
%o A381462 print(l)
%Y A381462 Cf. A376116, A381463.
%K A381462 nonn
%O A381462 1,3
%A A381462 _Ryota Inagaki_, _Tanya Khovanova_, and _Austin Luo_, Feb 24 2025