A332673 - OEIS

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.

%I #8 Feb 17 2023 11:49:36

%S 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,

%T 0,1,126,315,438,323,128,21,0,1,254,762,1305,1270,747,243,34,0,1,510,

%U 1785,3612,4346,3370,1629,452,55

%N 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.

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

%e Triangle begins:

%e 1

%e 0 1

%e 0 1 2

%e 0 1 6 3

%e 0 1 14 14 5

%e 0 1 30 45 32 8

%e 0 1 62 124 131 65 13

%e 0 1 126 315 438 323 128 21

%e 0 1 254 762 1305 1270 747 243 34

%e ...

%e Row n = 4 counts the following ordered set partitions:

%e {1234} {1}{234} {1}{2}{34} {1}{2}{3}{4}

%e {12}{34} {1}{23}{4} {1}{2}{4}{3}

%e {123}{4} {12}{3}{4} {1}{3}{2}{4}

%e {124}{3} {1}{24}{3} {2}{1}{3}{4}

%e {13}{24} {12}{4}{3} {2}{1}{4}{3}

%e {134}{2} {1}{3}{24}

%e {14}{23} {13}{2}{4}

%e {2}{134} {1}{34}{2}

%e {23}{14} {1}{4}{23}

%e {234}{1} {2}{1}{34}

%e {24}{13} {2}{13}{4}

%e {3}{124} {2}{14}{3}

%e {34}{12} {23}{1}{4}

%e {4}{123} {3}{12}{4}

%t sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];

%t Table[Length[Select[Join@@Permutations/@sps[Range[n]],Length[#]==k&&!MatchQ[#,{___,{___,a_,___},__,{___,b_,___},___}/;a>b]&]],{n,0,5},{k,0,n}]

%Y An apparently related triangle is A056242.

%Y Column k = n - 1 is A332724.

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

%Y Ordered set-partitions are A000670.

%Y Unimodal compositions are A001523.

%Y Non-unimodal normal sequences are A328509.

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

%K nonn,tabl

%O 0,6

%A _Gus Wiseman_, Mar 02 2020