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

0,2

Comments

Rows can be extended to negative k with T(n, -k) = T(n, k). Sums of such extended rows give A002894.

T(n, k) is the number of words of length 2n equal to z^k in the Heisenberg group, presented as <x, y | [x,z], [y,z]>, where z=[x,y]. In particular, T(n, 0) = A307468(n).

Links

Table of n, a(n) for n=0..41.

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.

Example

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.

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}]

Crossrefs

Row lengths are A033638 = A002620 + 1.

Row n ends with 4 * A026741(n) for n > 0.

Row 16 is A178106.

Cf. A307468, A002894, A353091.

A334756 counts self-avoiding walks only.

Sequence in context: A307031 A306823 A174212 * A038706 A358863 A222594

Adjacent sequences: A352835 A352836 A352837 * A352839 A352840 A352841

Keyword

nonn,tabf,walk

Author

Andrey Zabolotskiy, Apr 05 2022

Status

approved