A071157 -id:A071157 - cp's OEIS frontend

A071153 Łukasiewicz word for each rooted plane tree (interpretation e in Stanley's exercise 19) encoded by A014486 (or A063171), with the last leaf implicit, i.e., these words are given without the last trailing zero, except for the null tree which is encoded as 0.

0, 1, 20, 11, 300, 201, 210, 120, 111, 4000, 3001, 3010, 2020, 2011, 3100, 2101, 2200, 1300, 1201, 2110, 1210, 1120, 1111, 50000, 40001, 40010, 30020, 30011, 40100, 30101, 30200, 20300, 20201, 30110, 20210, 20120, 20111, 41000, 31001, 31010

Offset: 0

Antti Karttunen, May 14 2002

Note: this finite decimal representation works only up to the 6917th term, as the 6918th such word is already (10,0,0,0,0,0,0,0,0,0). The sequence A071154 shows the initial portion of this sequence sorted.

			The 11th term of A063171 is 10110010, corresponding to parenthesization ()(())(), thus its Łukasiewicz word is 3010. The 18th term of A063171 is 11011000, corresponding to parenthesization (()(())), thus its Łukasiewicz word is 1201. I.e., in the latter example there is one list on the top-level, which in turn contains two sublists, of which the first is zero elements long and the second is a sublist containing one empty sublist (the last zero is omitted).

A. Karttunen, Gatomorphisms and other excursions amidst the plane trees and parenthesizations (Includes the complete Scheme program for computing this sequence)
R. P. Stanley, Hipparchus, Plutarch, Schröder and Hough, Am. Math. Monthly, Vol. 104, No. 4, p. 344, 1997.
R. P. Stanley, Exercises on Catalan and Related Numbers
OEIS Wiki, Łukasiewicz words
Index entries for sequences related to Łukasiewicz
Index entries for sequences related to parenthesizing

For n >= 1, the number of zeros in the term a(n) is given by A057514(n)-1.
The first digit of each term is given by A057515.
Cf. A014486, A059984, A059985, A071152, A071154.
Corresponding factorial walk encoding: A071155 (A071157, A071159).
a(n) = A079436(n)/10.

A071159 Integers whose decimal expansion start with 1, do not contain zeros and each successive digit to the right is at most one greater than the previous digit.

Original entry on oeis.org

1, 11, 12, 111, 112, 121, 122, 123, 1111, 1112, 1121, 1122, 1123, 1211, 1212, 1221, 1222, 1223, 1231, 1232, 1233, 1234, 11111, 11112, 11121, 11122, 11123, 11211, 11212, 11221, 11222, 11223, 11231, 11232, 11233, 11234, 12111, 12112, 12121

Offset: 1

Antti Karttunen, May 14 2002

Robert Israel, Table of n, a(n) for n = 1..10000
S. Giraudo, Combinatorial operads from monoids, arXiv preprint arXiv:1306.6938 [math.CO], 2013-2015. See Sect. 3.1.3.

Essentially the same as A071157 but with digits reversed.

Corresponding Łukasiewicz words: A071153.

R[1]:= [1]:
for d from 2 to 6 do
R[d]:= map(t -> seq(10*t+j,j=1..min((t mod 10)+1,9)), R[d-1])
od:
A:= map(op, [seq(R[d],d=1..6)]); # Robert Israel, Jan 31 2017

desQ[n_]:=Module[{idn=IntegerDigits[n]},idn[[1]]==1&&FreeQ[idn,0]&&Max[ Differences[ idn]]<2]; Select[Range[13000],desQ] (* Harvey P. Dale, Feb 19 2017 *)

A071155 The Catalan factorial walks (for each rooted plane tree encoded by A014486) encoded as zero-free numbers in factorial base (A007623).

Original entry on oeis.org

0, 1, 3, 5, 9, 15, 11, 17, 23, 33, 57, 39, 63, 87, 35, 59, 41, 65, 89, 47, 71, 95, 119, 153, 273, 177, 297, 417, 159, 279, 183, 303, 423, 207, 327, 447, 567, 155, 275, 179, 299, 419, 161, 281, 185, 305, 425, 209, 329, 449, 569, 167, 287, 191, 311, 431, 215, 335

Offset: 0

Antti Karttunen, May 14 2002

C. Banderier, A. Denise, P. Flajolet, M. Bousquet-Mélou, et al., Generating Functions for Generating Trees, Discrete Mathematics 246(1-3), March 2002, pp. 29-55.
A. Karttunen, Gatomorphisms and other excursions amidst the plane trees and parenthesizations (Includes the complete Scheme program for computing this sequence)

Same sequence sorted: A071156, expanded in the factorial number system: A071157. Corresponding Łukasiewicz words: A071153.

Cf. A000108 (row lengths), A120695.

A071158 Factorial expansion of A071156.

Original entry on oeis.org

1, 11, 21, 111, 121, 211, 221, 321, 1111, 1121, 1211, 1221, 1321, 2111, 2121, 2211, 2221, 2321, 3211, 3221, 3321, 4321, 11111, 11121, 11211, 11221, 11321, 12111, 12121, 12211, 12221, 12321, 13211, 13221, 13321, 14321, 21111, 21121, 21211

Offset: 1

Antti Karttunen, May 14 2002

Note that this decimal representation works only as long as there does not appear any digit 'ten' or higher in the factorial expansion of A071156.

A. Karttunen, Illustration of 626 initial terms (up to seven digits) showing a simple bijection with non-crossing Murasaki-diagrams

a(n) = A007623(A071156(n)) = A071157(A057164(n)).

cp's OEIS Frontend

A071153 Łukasiewicz word for each rooted plane tree (interpretation e in Stanley's exercise 19) encoded by A014486 (or A063171), with the last leaf implicit, i.e., these words are given without the last trailing zero, except for the null tree which is encoded as 0.

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A071159 Integers whose decimal expansion start with 1, do not contain zeros and each successive digit to the right is at most one greater than the previous digit.

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Mathematica

A071155 The Catalan factorial walks (for each rooted plane tree encoded by A014486) encoded as zero-free numbers in factorial base (A007623).

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A071158 Factorial expansion of A071156.

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