

A334440


Irregular triangle T(n,k) read by rows: row n lists numbers of distinct parts of the nth integer partition in AbramowitzStegun (sum/length/lex) order.


15



0, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 2, 2, 2, 2, 1, 1, 1, 2, 2, 1, 3, 2, 2, 2, 2, 1, 1, 2, 2, 2, 2, 2, 3, 2, 2, 3, 2, 2, 2, 2, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 2, 1, 3, 2, 3, 2, 2, 3, 2, 2, 2, 2, 1, 1, 2, 2, 2, 2, 1, 3, 2, 2, 3, 3, 2, 2, 3, 3, 3, 3, 2, 2, 3
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OFFSET

0,6


COMMENTS

The total number of parts, counting duplicates, is A036043. The version for reversed partitions is A103921.


LINKS

Robert Price, Table of n, a(n) for n = 0..9295 (25 rows).
Wikiversity, Lexicographic and colexicographic order


FORMULA

a(n) = A001221(A334433(n)).


EXAMPLE

Triangle begins:
0
1
1 1
1 2 1
1 1 2 2 1
1 2 2 2 2 2 1
1 1 2 2 1 3 2 2 2 2 1
1 2 2 2 2 2 3 2 2 3 2 2 2 2 1
1 1 2 2 2 2 2 3 3 2 1 3 2 3 2 2 3 2 2 2 2 1


MATHEMATICA

Join@@Table[Length/@Union/@Sort[IntegerPartitions[n]], {n, 0, 10}]


CROSSREFS

Row lengths are A000041.
The number of not necessarily distinct parts is A036043.
The version for reversed partitions is A103921.
Ignoring length (sum/lex) gives A103921 (also).
a(n) is the number of distinct elements in row n of A334301.
The maximum part of the same partition is A334441.
Lexicographically ordered reversed partitions are A026791.
Reversed partitions in AbramowitzStegun (sum/length/lex) order are A036036.
Partitions in increasinglength colex order (sum/length/colex) are A036037.
Graded reverselexicographically ordered partitions are A080577.
Partitions counted by sum and number of distinct parts are A116608.
Graded lexicographically ordered partitions are A193073.
Partitions in colexicographic order (sum/colex) are A211992.
Partitions in dual AbramowitzStegun (sum/length/revlex) order are A334439.
Cf. A001221, A049085, A124734, A185974, A296774, A299755, A334028, A334302, A334433, A334434, A334435, A334438.
Sequence in context: A355241 A165190 A025890 * A316975 A043277 A265991
Adjacent sequences: A334437 A334438 A334439 * A334441 A334442 A334443


KEYWORD

nonn,tabf


AUTHOR

Gus Wiseman, May 05 2020


STATUS

approved



