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

1,2

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

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 *)

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.

Sequence in context: A271355 A211230 A049085 * A331581 A227355 A226080

Adjacent sequences: A193170 A193171 A193172 * A193174 A193175 A193176

Keyword

nonn,look,tabf

Author

Alois P. Heinz, Jul 17 2011

Status

approved