A240763 Preferential arrangements of 1, 2, 3, ... things in one-line notation, arranged lexicographically.

1, 11, 12, 21, 111, 112, 121, 122, 123, 132, 211, 212, 213, 221, 231, 312, 321, 1111, 1112, 1121, 1122, 1123, 1132, 1211, 1212, 1213, 1221, 1222, 1223, 1231, 1232, 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

Offset: 1

Author

N. J. A. Sloane, Apr 12 2014

Keywords

• nonn

Comments

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

There are A000670(n) terms of length n.

Examples

			The preferential arrangement of 7 things given by
3=4 < 5 < 1=2=7 < 6
would be represented by
1 2 3 4 5 6 7
3 3 1 1 2 4 3
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.
Thanks to _Nathaniel Shar_ for suggesting the one-line notation.
		

Links

• N. J. A. Sloane, Table of n, a(n) for n = 1..52609 (lists all preferential arrangements of <= 7 things).
• N. J. A. Sloane, List of preferential arrangements on 1 thru 5 things, in human-readable notation [These are in a different order from those in the b-file]

Crossrefs

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