A191385 -id:A191385 - cp's OEIS frontend

A191384 Triangle read by rows: T(n,k) is the number of dispersed Dyck paths of length n (i.e., Motzkin paths of length n with no (1,0) steps at positive heights) with k ascents of length 1. An ascent is a maximal sequence of consecutive (1,1)-steps.

1, 1, 1, 1, 1, 2, 2, 3, 1, 3, 4, 3, 5, 8, 6, 1, 7, 14, 10, 4, 12, 26, 21, 10, 1, 18, 42, 41, 20, 5, 31, 77, 83, 45, 15, 1, 47, 128, 150, 96, 35, 6, 81, 234, 293, 209, 85, 21, 1, 125, 388, 530, 414, 196, 56, 7, 216, 704, 1023, 858, 455, 147, 28, 1, 337, 1172, 1828, 1668, 974, 364, 84, 8, 583, 2119, 3479, 3385, 2133, 896, 238, 36, 1

Offset: 0

Emeric Deutsch, Jun 01 2011

Row n has 1 + floor(n/2) entries.
Sum of entries in row n is binomial(n, floor(n/2)) = A001405(n).
T(n,0) = A191385(n).
Sum_{k>=0} k*T(n,k) = A191386(n).

			T(5,2)=3 because we have HUDUD, UDHUD, and UDUDH, where U=(1,1), D=(1,-1), H=(1,0).
Triangle starts:
  1;
  1;
  1,  1;
  1,  2;
  2,  3,  1;
  3,  4,  3;
  5,  8,  6,  1;
  7, 14, 10,  4;

Cf. A001405, A191385, A191386.

G := ((t*z^2-(1-z)^2+sqrt((1+z^2-t*z^2)*(1-3*z^2-t*z^2)))*1/2)/(z*(1-2*z+z^2-z^3-t*z^2+t*z^3)): Gser := simplify(series(G, z = 0, 19)): for n from 0 to 16 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 0 to 16 do seq(coeff(P[n], t, k), k = 0 .. floor((1/2)*n)) end do; # yields sequence in triangular form

G.f.: G(t,z) = (t*z^2 - (1-z)^2 + sqrt((1+z^2-t*z^2)*(1-3*z^2-t*z^2)))/(2*z*(1-2*z+z^2-z^3-t*z^2+t*z^3)).

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.

Original entry on oeis.org

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

Sergey Kirgizov, Apr 08 2018

nonn

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.

			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}.

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

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.

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 *)

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

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).

cp's OEIS Frontend

A191384 Triangle read by rows: T(n,k) is the number of dispersed Dyck paths of length n (i.e., Motzkin paths of length n with no (1,0) steps at positive heights) with k ascents of length 1. An ascent is a maximal sequence of consecutive (1,1)-steps.

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