cp's OEIS Frontend

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.

%I A381463 #35 Jul 23 2025 16:07:26
%S A381463 0,1,4,5,6,16,17,20,21,22,24,25,26,27,64,65,68,69,70,80,81,84,85,86,
%T A381463 88,89,90,91,96,97,100,101,102,104,105,106,107,108,109,110,111,112,
%U A381463 256,257,260,261,262,272,273,276,277,278,280,281,282,283,320,321,324,325,326,336,337,340
%N 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.
%C A381463 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.
%C A381463 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.
%C A381463 This sequence was defined at the end of Section 4.3 of Inagaki, Khovanova, and Luo (2025).
%H A381463 Wikipedia, <a href="https://en.wikipedia.org/wiki/Chip-firing_game">Chip-firing game</a>
%H A381463 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 A381463 (Python)
%o A381463 k = 4
%o A381463 s = set()
%o A381463 for i in range(2):
%o A381463     for j in range(3):
%o A381463         for l in range(4):
%o A381463             for m in range(5):
%o A381463                 for n in range(6):
%o A381463                     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 A381463 l = list(s)
%o A381463 l.sort()
%o A381463 print(l)
%Y A381463 Cf. A376116, A381462.
%K A381463 nonn
%O A381463 1,3
%A A381463 _Ryota Inagaki_, _Tanya Khovanova_, and _Austin Luo_, Feb 24 2025