

A193173


Triangle in which nth row lists the number of elements in lexicographically ordered partitions of n, A026791.


12



1, 2, 1, 3, 2, 1, 4, 3, 2, 2, 1, 5, 4, 3, 3, 2, 2, 1, 6, 5, 4, 4, 3, 3, 2, 3, 2, 2, 1, 7, 6, 5, 5, 4, 4, 3, 4, 3, 3, 2, 3, 2, 2, 1, 8, 7, 6, 6, 5, 5, 4, 5, 4, 4, 3, 4, 3, 3, 2, 4, 3, 3, 2, 2, 2, 1, 9, 8, 7, 7, 6, 6, 5, 6, 5, 5, 4, 5, 4, 4, 3, 5, 4, 4, 3, 3, 3, 2, 4, 3, 3, 2, 3, 2, 2, 1, 10, 9, 8, 8, 7, 7, 6, 7, 6
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OFFSET

1,2


COMMENTS

This sequence first differs from A049085 in the partitions of 6 (at flattened index 22):
6, 5, 4, 4, 3, 3, 2, 3, 2, 2, 1 (this sequence);
6, 5, 4, 3, 4, 3, 2, 3, 2, 2, 1 (A049085).
The name is correct if the partitions are read in reverse, so that the parts are weakly increasing. The version for nonreversed partitions is A049085.


LINKS



EXAMPLE

The lexicographically ordered partitions of 3 are [[1, 1, 1], [1, 2], [3]], thus row 3 has 3, 2, 1.
Triangle begins:
1;
2, 1;
3, 2, 1;
4, 3, 2, 2, 1;
5, 4, 3, 3, 2, 2, 1;
6, 5, 4, 4, 3, 3, 2, 3, 2, 2, 1;
...


MAPLE

T:= proc(n) local b, ll;
b:= proc(n, l)
if n=0 then ll:= ll, nops(l)
else seq(b(ni, [l[], i]), i=`if`(l=[], 1, l[1])..n) fi
end;
ll:= NULL; b(n, []); ll
end:
seq(T(n), n=1..11);


MATHEMATICA

lexsort[f_, c_]:=OrderedQ[PadRight[{f, c}]];
Table[Length/@Sort[Reverse/@IntegerPartitions[n], lexsort], {n, 0, 10}] (* Gus Wiseman, May 22 2020 *)


CROSSREFS

The version ignoring length is A036043.
The version for nonreversed partitions is A049085.
The maxima of these partitions are A194546.
Reversed partitions in AbramowitzStegun order are A036036.
Reverselexicographically ordered partitions are A080577.
Cf. A001222, A115623, A129129, A185974, A193073, A211992, A228531, A334302, A334434, A334437, A334440, A334441.


KEYWORD



AUTHOR



STATUS

approved



