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Preferential arrangements of 1, 2, 3, ... things in one-line notation, arranged lexicographically.
13

%I #23 Apr 13 2014 12:35:56

%S 1,11,12,21,111,112,121,122,123,132,211,212,213,221,231,312,321,1111,

%T 1112,1121,1122,1123,1132,1211,1212,1213,1221,1222,1223,1231,1232,

%U 1233,1234,1243,1312,1321,1322,1323,1324,1332,1342,1423,1432,2111,2112,2113,2121,2122,2123,2131,2132,2133,2134,2143,2211,2212

%N Preferential arrangements of 1, 2, 3, ... things in one-line notation, arranged lexicographically.

%C A preferential arrangement is like a permutation, except that ties are allowed. Preferential arrangements are also called ordered partitions. See A000670.

%C There are A000670(n) terms of length n.

%H N. J. A. Sloane, <a href="/A240763/b240763.txt">Table of n, a(n) for n = 1..52609</a> (lists all preferential arrangements of <= 7 things).

%H N. J. A. Sloane, <a href="/A240763/a240763.txt">List of preferential arrangements on 1 thru 5 things, in human-readable notation</a> [These are in a different order from those in the b-file]

%e The preferential arrangement of 7 things given by

%e 3=4 < 5 < 1=2=7 < 6

%e would be represented by

%e 1 2 3 4 5 6 7

%e 3 3 1 1 2 4 3

%e which in the compressed one-line notation is written 3311243. Obviously this compressed notation only works for fewer than 10 things. In the "human-readable" notation used in the a-file, this example would be written 34,5,127,6.

%e Thanks to _Nathaniel Shar_ for suggesting the one-line notation.

%Y Cf. A000670, A239914, A217389, A030299 (an analogous sequence for permutations).

%K nonn

%O 1,2

%A _N. J. A. Sloane_, Apr 12 2014