A352838 Irregular triangle read by rows: T(n, k) is the number of 2n-step closed walks on the square lattice having algebraic area k; n >= 0, 0 <= k <= floor(n^2/4).
1, 4, 28, 4, 232, 72, 12, 2156, 1008, 308, 48, 8, 21944, 13160, 5540, 1560, 420, 80, 20, 240280, 168780, 87192, 33628, 11964, 3636, 1200, 264, 72, 12, 2787320, 2168544, 1291220, 610232, 262612, 101976, 40376, 13720, 4900, 1512, 420, 112, 28
Offset: 0
Examples
The table begins:
1
4
28, 4
232, 72, 12
2156, 1008, 308, 48, 8
21944, 13160, 5540, 1560, 420, 80, 20
240280, 168780, 87192, 33628, 11964, 3636, 1200, 264, 72, 12
...
T(2, 0) = 28: the 4-step walks enclosing algebraic area 0 include 16 walks of the form "some two steps, then two steps right back" and 12 walks of the form "some step, step back, a different step, step back".
T(2, 1) = 4: the 4-step walks enclosing algebraic area 1 are the walks around each of the 4 squares touching the origin in the positive direction; cf. A334756(2, 1) = 8, which also counts walks around these squares in the negative direction.
Links
- Andrei Zabolotskii, Table of n, a(n) for n = 0..1403 (rows 0..25)
- Cédric Béguin, Alain Valette and Andrzej Zuk, On the spectrum of a random walk on the discrete Heisenberg group and the norm of Harper's operator, Journal of Geometry and Physics, 21 (1997), 337-356.
- Li Gan, Algebraic Area of Lattice Random Walks and Exclusion Statistics, PhD thesis, Université Paris-Saclay, 2023. See Section 2.1.2, in particular Table 2.1 (divide terms in rows with nonzero A by 2 to get this table).
- Stefan Mashkevich and Stéphane Ouvry, Area Distribution of Two-Dimensional Random Walks on a Square Lattice, J. Stat. Phys., 137 (2009), 71-78.
- Morteza Mohammad-Noori, Enumeration of closed random walks in the square lattice according to their areas, arXiv:1012.3720 [math.CO], 2010. Published as: Morteza Mohammad-Noori, Enumeration of walks in the square lattice according to their areas, Journal of Combinatorial Mathematics and Combinatorial Computing, 91 (2014), 257-274.
Crossrefs
Programs
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Mathematica
z[0, 0, 0, 0] = 1; z[-1, ] = z[, -1, _] = z[, , -1, ] = z[, , , -1] = 0; z[m1_, m2_, l1_, l2_] := z[m1, m2, l1, l2] = Expand[z[m1, m2, l1 - 1, l2] + z[m1, m2, l1, l2 - 1] + q^(l2 - l1) z[m1 - 1, m2, l1, l2] + q^(l1 - l2) z[m1, m2 - 1, l1, l2]]; zN[n_] := Sum[z[m, m, n/2 - m, n/2 - m], {m, 0, n/2}]; walks[n_] := Module[{gf = zN[2 n], e}, e = Exponent[gf, q, Max]; CoefficientList[gf q^e, q][[e + 1 ;;]]]; Table[walks[n], {n, 0, 8}]
Comments