A332673 Triangle read by rows where T(n,k) is the number of length-k ordered set partitions of {1..n} whose non-adjacent blocks are pairwise increasing.

1, 0, 1, 0, 1, 2, 0, 1, 6, 3, 0, 1, 14, 14, 5, 0, 1, 30, 45, 32, 8, 0, 1, 62, 124, 131, 65, 13, 0, 1, 126, 315, 438, 323, 128, 21, 0, 1, 254, 762, 1305, 1270, 747, 243, 34, 0, 1, 510, 1785, 3612, 4346, 3370, 1629, 452, 55

Offset

0,6

Comments

In other words, parts of subsequent, non-successive blocks are increasing.

Links

Table of n, a(n) for n=0..54.

Example

Triangle begins:
    1
    0    1
    0    1    2
    0    1    6    3
    0    1   14   14    5
    0    1   30   45   32    8
    0    1   62  124  131   65   13
    0    1  126  315  438  323  128   21
    0    1  254  762 1305 1270  747  243   34
    ...
Row n = 4 counts the following ordered set partitions:
  {1234}  {1}{234}  {1}{2}{34}  {1}{2}{3}{4}
          {12}{34}  {1}{23}{4}  {1}{2}{4}{3}
          {123}{4}  {12}{3}{4}  {1}{3}{2}{4}
          {124}{3}  {1}{24}{3}  {2}{1}{3}{4}
          {13}{24}  {12}{4}{3}  {2}{1}{4}{3}
          {134}{2}  {1}{3}{24}
          {14}{23}  {13}{2}{4}
          {2}{134}  {1}{34}{2}
          {23}{14}  {1}{4}{23}
          {234}{1}  {2}{1}{34}
          {24}{13}  {2}{13}{4}
          {3}{124}  {2}{14}{3}
          {34}{12}  {23}{1}{4}
          {4}{123}  {3}{12}{4}

Mathematica

sps[{}]:={{}}; sps[set:{i_, ___}]:=Join@@Function[s, Prepend[#, s]&/@sps[Complement[set, s]]]/@Cases[Subsets[set], {i, ___}];
Table[Length[Select[Join@@Permutations/@sps[Range[n]], Length[#]==k&&!MatchQ[#, {___, {___, a_, ___}, __, {___, b_, ___}, ___}/; a>b]&]], {n, 0, 5}, {k, 0, n}]

Crossrefs

An apparently related triangle is A056242.

Column k = n - 1 is A332724.

Row sums are A332872, which appears to be A007052 shifted right once.

Ordered set-partitions are A000670.

Unimodal compositions are A001523.

Non-unimodal normal sequences are A328509.

Cf. A072704, A097805, A107429, A227038, A332280, A332283, A332288, A332577.

Sequence in context: A367270 A365770 A059299 * A128722 A324659 A304485

Adjacent sequences: A332670 A332671 A332672 * A332674 A332675 A332676

Keyword

nonn,tabl

Author

Gus Wiseman, Mar 02 2020

Status

approved