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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.

%I A332673 #8 Feb 17 2023 11:49:36
%S A332673 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 A332673 0,1,126,315,438,323,128,21,0,1,254,762,1305,1270,747,243,34,0,1,510,
%U A332673 1785,3612,4346,3370,1629,452,55
%N 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.
%C A332673 In other words, parts of subsequent, non-successive blocks are increasing.
%e A332673 Triangle begins:
%e A332673     1
%e A332673     0    1
%e A332673     0    1    2
%e A332673     0    1    6    3
%e A332673     0    1   14   14    5
%e A332673     0    1   30   45   32    8
%e A332673     0    1   62  124  131   65   13
%e A332673     0    1  126  315  438  323  128   21
%e A332673     0    1  254  762 1305 1270  747  243   34
%e A332673     ...
%e A332673 Row n = 4 counts the following ordered set partitions:
%e A332673   {1234}  {1}{234}  {1}{2}{34}  {1}{2}{3}{4}
%e A332673           {12}{34}  {1}{23}{4}  {1}{2}{4}{3}
%e A332673           {123}{4}  {12}{3}{4}  {1}{3}{2}{4}
%e A332673           {124}{3}  {1}{24}{3}  {2}{1}{3}{4}
%e A332673           {13}{24}  {12}{4}{3}  {2}{1}{4}{3}
%e A332673           {134}{2}  {1}{3}{24}
%e A332673           {14}{23}  {13}{2}{4}
%e A332673           {2}{134}  {1}{34}{2}
%e A332673           {23}{14}  {1}{4}{23}
%e A332673           {234}{1}  {2}{1}{34}
%e A332673           {24}{13}  {2}{13}{4}
%e A332673           {3}{124}  {2}{14}{3}
%e A332673           {34}{12}  {23}{1}{4}
%e A332673           {4}{123}  {3}{12}{4}
%t A332673 sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t A332673 Table[Length[Select[Join@@Permutations/@sps[Range[n]],Length[#]==k&&!MatchQ[#,{___,{___,a_,___},__,{___,b_,___},___}/;a>b]&]],{n,0,5},{k,0,n}]
%Y A332673 An apparently related triangle is A056242.
%Y A332673 Column k = n - 1 is A332724.
%Y A332673 Row sums are A332872, which appears to be A007052 shifted right once.
%Y A332673 Ordered set-partitions are A000670.
%Y A332673 Unimodal compositions are A001523.
%Y A332673 Non-unimodal normal sequences are A328509.
%Y A332673 Cf. A072704, A097805, A107429, A227038, A332280, A332283, A332288, A332577.
%K A332673 nonn,tabl
%O A332673 0,6
%A A332673 _Gus Wiseman_, Mar 02 2020