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).
- Jason Kimberley, Oct 27 2011
Rows sums give A006128, n >= 1. - Omar E. Pol, Dec 06 2011
The name is correct if the partitions are read in reverse, so that the parts are weakly increasing. The version for non-reversed partitions is A049085.
LINKS
Alois P. Heinz, Rows n = 1..26, flattened
FORMULA
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(n-i, [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 *)
PROG
(PARI) Row(n)=[#p | p<-vecsort(partitions(n))]
{ for(n=0, 9, print(Row(n))) } \\ Andrew Howroyd, Oct 06 2025
CROSSREFS
Row lengths are A000041.
Partition lengths of A026791.
The version ignoring length is A036043.
The version for non-reversed partitions is A049085.
The maxima of these partitions are A194546.
Reversed partitions in Abramowitz-Stegun order are A036036.
Reverse-lexicographically ordered partitions are A080577.
KEYWORD
AUTHOR
Alois P. Heinz, Jul 17 2011
STATUS
approved
