A193173 Triangle in which n-th row lists the number of elements in lexicographically ordered partitions of n, A026791.

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

Offset: 1

Author

Alois P. Heinz, Jul 17 2011

Keywords

• nonn
• look
• tabf

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.

Examples

			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;
  ...
		

Links

• Alois P. Heinz, Rows n = 1..26, flattened

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.

Cf. A001222, A115623, A129129, A185974, A193073, A211992, A228531, A334302, A334434, A334437, A334440, A334441.

Programs

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 *)