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Number of ways to partition the Young diagram of an integer partition of n into vertical sections.
7

%I #12 Aug 29 2023 13:12:43

%S 1,1,3,9,37,152,780,3965,23460,141471,944217,6445643,48075092,

%T 364921557,2974423953,24847873439,219611194148,1987556951714,

%U 18930298888792,184244039718755,1874490999743203,19510832177784098,210941659716920257,2331530519337226199,26692555830628617358

%N Number of ways to partition the Young diagram of an integer partition of n into vertical sections.

%C A vertical section is a partial Young diagram with at most one square in each row. For example, a partition (shown as a coloring by positive integers) into vertical sections of the Young diagram of (322) is:

%C 1 2 3

%C 1 2

%C 2 3

%e The a(4) = 37 partitions into vertical sections of integer partitions of 4:

%e 1 2 3 4

%e .

%e 1 2 3 1 2 3 1 2 3 1 2 3

%e 4 3 2 1

%e .

%e 1 2 1 2 1 2 1 2 1 2 1 2 1 2

%e 3 4 2 3 3 2 1 3 1 2 3 1 2 1

%e .

%e 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2

%e 3 3 2 3 2 1 1 3 2 1

%e 4 3 3 2 2 3 2 1 1 1

%e .

%e 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

%e 2 2 2 2 2 1 1 2 2 2 2 1 1 2 1

%e 3 3 2 3 2 2 2 1 1 3 2 1 2 1 1

%e 4 3 3 2 2 3 2 3 2 1 1 2 1 1 1

%t spsu[_,{}]:={{}};spsu[foo_,set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@spsu[Select[foo,Complement[#,Complement[set,s]]=={}&],Complement[set,s]]]/@Cases[foo,{i,___}];

%t ptnpos[y_]:=Position[Table[1,{#}]&/@y,1];

%t ptnverts[y_]:=Select[Rest[Subsets[ptnpos[y]]],UnsameQ@@First/@#&];

%t Table[Sum[Length[spsu[ptnverts[y],ptnpos[y]]],{y,IntegerPartitions[n]}],{n,6}]

%Y Cf. A000110, A000258, A008277, A046682, A122111, A318396, A321728, A321729, A321730, A321731, A321738, A321854.

%K nonn

%O 0,3

%A _Gus Wiseman_, Nov 19 2018

%E a(11)-a(24) from _Ludovic Schwob_, Aug 28 2023