A334720
Number of 2-dimensional closed-loop self-avoiding paths on a square lattice where each path consists of steps with incrementing length from 1 to n.
Original entry on oeis.org
0, 0, 0, 0, 0, 0, 8, 24, 0, 0, 40, 112, 0, 0, 1376, 2008, 0, 0, 21720, 60848, 0, 0, 635544, 1517368, 0, 0, 20008456, 46010640, 0, 0, 640819936, 1571759136, 0, 0, 22704325648, 55436103264
Offset: 1
a(1) to a(6) = 0 as no closed-loop is possible.
a(7) = 8 as there is one path which forms a closed loop which can be walked in 8 different ways on a 2D square lattice. The path is:
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5
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. . 4
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6 . .
| | 3
. *---.---.---*
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. . 2
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7 1
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See the attached link for text images of the closed loops for other n values.
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).
Original entry on oeis.org
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
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.
- 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.
Row n ends with 4 *
A026741(n) for n > 0.
A334756 counts self-avoiding walks only.
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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}]
A335305
Number of 2-dimensional closed-loop self-avoiding paths on a square lattice where each path consists of steps of length 1 to n which can be taken in any order.
Original entry on oeis.org
0, 0, 0, 0, 0, 0, 16128, 287232, 0, 0, 1843367680, 45589291776, 0, 0
Offset: 1
a(1) to a(6) = 0 as no closed loop paths are possible.
a(7) = 16128. Given the first step has length 1 and is to the right, with the next non-right step being upward, there are 84 different loops. Each of these can be walked in at least 2 ways, with the single perfect square having 48 different possible walks. Each of these in turn can be started with a first step of length 1 to n, and each of these can then be walked in 8 different ways on a 2D square grid. This gives a total number of 7-step paths of 16128. This should be compared with A334720 where for n=7 only 8 paths are possible. See the attached link text file for more details of n=7.
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