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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.
6
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.
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.
Sequence in context: A367270 A365770 A059299 * A128722 A324659 A304485
KEYWORD
nonn,tabl
AUTHOR
Gus Wiseman, Mar 02 2020
STATUS
approved