A278985 List of words of length n over an alphabet of size 3 that are in standard order.

1, 11, 12, 111, 112, 121, 122, 123, 1111, 1112, 1121, 1122, 1123, 1211, 1212, 1213, 1221, 1222, 1223, 1231, 1232, 1233, 11111, 11112, 11121, 11122, 11123, 11211, 11212, 11213, 11221, 11222, 11223, 11231, 11232, 11233, 12111, 12112, 12113, 12121, 12122

Offset: 1

Author

N. J. A. Sloane, Dec 05 2016

Keywords

• nonn
• base

Comments

We study words made of letters from an alphabet of size b, where b >= 1. We assume the letters are labeled {1,2,3,...,b}. There are b^n possible words of length n.

We say that a word is in "standard order" if it has the property that whenever a letter i appears, the letter i-1 has already appeared in the word. This implies that all words begin with the letter 1.

These are the words described in row b=3 of the array in A278984.

A007051(n-1) gives the number of n-digit terms in this sequence. - Rémy Sigrist, Dec 18 2016

Links

• Rémy Sigrist and N. J. A. Sloane, Table of n, a(n) for n = 1..14767 [Terms 1 through 185 by N. J. A. Sloane]
• Joerg Arndt and N. J. A. Sloane, Counting Words that are in "Standard Order"

Crossrefs

Cf. A278984, A007051.

Similar to but different from A071159.

Programs

Maple

b:= proc(n) option remember; `if`(n=1, [[1]], map(x->
      seq([x[], i], i=1..min(3, max(x[])+1)), b(n-1)))
    end:
T:= n-> map(x-> parse(cat(x[])), b(n))[]:
seq(T(n), n=1..5);  # Alois P. Heinz, Jan 02 2022

Mathematica

Table[FromDigits /@ Select[Tuples[Range@ 3, n], And[Times @@ Boole@ MapIndexed[#1 <= First@ #2 &, #] > 0, Max@ Differences@ # <= 1] &], {n, 5}] // Flatten (* Michael De Vlieger, Dec 18 2016 *)

PARI

gen(n, len, mx) = if (len==0, print1 (n ", "), for (d=1, min(mx+1, 3), gen(10*n + d, len-1, max(mx, d))))
for (len=1, 5, gen(0, len, 0)) \\ Rémy Sigrist, Dec 18 2016