A348566 Triangle read by rows: T(m, n) is the number of symmetric recurrent sandpiles on an m X n grid (m >= 0, 0 <= n <= m).

1, 1, 4, 1, 3, 2, 1, 14, 7, 128, 1, 11, 5, 71, 36, 1, 52, 18, 1358, 539, 43264, 1, 41, 13, 769, 281, 17753, 6728, 1, 194, 47, 14852, 4271, 1452866, 434657, 151519232, 1, 153, 34, 8449, 2245, 603126, 167089, 46069729, 12988816, 1, 724, 123, 163534, 34276, 49704772, 10894561, 16236962114, 3625549353, 5475450241024

Offset: 0

Author

Andrey Zabolotskiy, Oct 22 2021

Keywords

• nonn
• tabl

Comments

Terms of this triangle count recurrent sandpiles on rectangular grids that have vertical and horizontal symmetries. Terms of A348567 count recurrent sandpiles on square grids that also have diagonal symmetries.

Examples

			The triangle begins:
  1
  1  4
  1  3  2
  1 14  7  128
  1 11  5   71  36
  1 52 18 1358 539 43264
  1 41 13  769 281 17753 6728
...
See Fig. 9 of the paper by Florescu et al. for the T(4, 4) = 36 symmetric recurrent sandpiles on a 4x4 grid.
		

Links

• Laura Florescu, Daniela Morar, David Perkinson, Nicholas Salter and Tianyuan Xu, Sandpiles and Dominos, El. J. Comb., 22 (2015), P1.66.
• Wikipedia, Abelian sandpile model

Crossrefs

Cf. A348567, A187617, A187618, A103997, A256045, A116469.

Formula

T(2m, 2n) = A187617(m, n) = A187618(m, n). [Florescu et al., Theorem 15]

T(2m, 2n-1) = T(2n-1, 2m) = A103997(m, n). [Florescu et al., Theorem 18]

T(2m-1, 2n-1) = Product_{h=1..m, k=1..n} 4*(z(h, m) + z(k, n)) where z(k, n) = cos(Pi*(2k-1)/(4n)). [Florescu et al., Theorem 23]

A256045(m, n) divides T(m, n), T(m, n) divides A116469(m+1, n+1).

This triangle can obviously be extended to n > m as T(m, n) = T(n, m).