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A193023 Triangle read by rows: the n-th row has length A000110(n) and contains all set partitions of an n-set in canonical order. 1
1, 11, 12, 111, 112, 121, 122, 123, 1111, 1112, 1121, 1122, 1123, 1211, 1212, 1213, 1221, 1222, 1223, 1231, 1232, 1233, 1234, 11111, 11112, 11121, 11122, 11123, 11211, 11212, 11213, 11221, 11222, 11223, 11231, 11232, 11233, 11234, 12111, 12112, 12113, 12121 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

The set partition of [1,2,3,4] given by 13/2/4 would be encoded as 1213: simply record which part i is in, for i=1..n.

To get row n, read row n-1 from left to right. If row n-1 contains a word abc...d, in which the maximal number is m, then in row n we place the words abc...d1, abc...d2, abc...d3, ..., abc...d(m+1).

This provides a canonical ordering for partitions of a labeled set.

LINKS

Alois P. Heinz, Rows n = 1..8, flattened

R. Kaye, A Gray code for set partitions, Info. Proc. Letts., 5 (1976), 171-173.

EXAMPLE

Triangle begins:

1;

11,12;

111,112,121,122,123;

1111,1112,1121,1122,1123,1211,1212,1213,1221,1222,1223,1231,1232,1233,1234;

11111,11112,11121,11122,11123,...

MAPLE

b:= proc(n) option remember;

      `if`(n=1, [[1]], map(x-> seq([x[], i], i=1..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, Sep 30 2011

CROSSREFS

This is different from A071159.

Sequence in context: A082262 A239463 A153070 * A278985 A071159 A231873

Adjacent sequences:  A193020 A193021 A193022 * A193024 A193025 A193026

KEYWORD

nonn,tabf

AUTHOR

N. J. A. Sloane, Jul 14 2011

STATUS

approved

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Last modified February 20 18:29 EST 2018. Contains 299381 sequences. (Running on oeis4.)