A302483 Number of FF-equivalence classes of Łukasiewicz paths. Łukasiewicz paths are P-equivalent iff the positions of pattern P are identical in these paths.

1, 1, 2, 2, 5, 9, 17, 32, 59, 107, 192, 342, 606, 1070, 1885, 3316, 5828, 10237, 17975, 31555, 55387, 97210, 170605, 299405, 525434, 922088, 1618168, 2839704, 4983351, 8745190, 15346758, 26931703, 47261865, 82938813, 145547493, 255418068, 448227487, 786584431

Offset: 0

Author

Sergey Kirgizov, Apr 08 2018

Keywords

• nonn

Comments

Number of FF-equivalence classes of Łukasiewicz paths. A Łukasiewicz path of length n is a lattice path from (0,0) to (n,0) using up steps U_{k} = (1,k) for any positive integer k, flat steps F = (1,0) and down steps D = (1,-1). Łukasiewicz paths are alpha-equivalent whenever the positions of occurrences of pattern alpha are identical on these paths.

Examples

			There are 14 Łukasiewicz of length 4 divided in the 5 following FF-equivalence classes: {FFFF}, {FFU_{1}D}, {U_{1}DFF}, {U_{1}FFD}, {FU_{1}DF, FU_{1}FD, FU_{2}DD, U_{1}DU_{1}D, U_{1}FDF, U_{1}U_{1}DD, U_{2}DDF, U_{2}DFD, U_{2}FDD, U_{3}DDD}.
		

Links

• Jean-Luc Baril, Sergey Kirgizov and Armen Petrossian, Enumeration of Łukasiewicz paths modulo some patterns, arXiv:1804.01293 [math.CO], 2018.

Crossrefs

Cf. A001405, A191385, A000045, A005251, A000325, A011782, A001006, A023431, A292460, A004148 enumerates the numbers of P-equivalence classes of Łukasiewicz paths for other values of P.

Programs

Mathematica

CoefficientList[Series[(1 - 3 x + 4 x^2 - 5 x^3 + 7 x^4 - 7 x^5 + 6 x^6 - 3 x^7 + x^8)/((1 - 2 x + x^2 - x^3) (1 - x)^2), {x, 0, 32}], x] (* Michael De Vlieger, Apr 12 2018 *)

PARI

x='x+O('x^99); Vec((1-3*x+4*x^2-5*x^3+7*x^4-7*x^5+6*x^6-3*x^7+x^8)/((1-2*x+x^2-x^3)*(1-x)^2)) \\ Altug Alkan, Apr 12 2018

Formula

G.f.: (1 - 3*x + 4*x^2 - 5*x^3 + 7*x^4 - 7*x^5 + 6*x^6 - 3*x^7 + x^8) / ((1-2*x+x^2-x^3) * (1-x)^2).