A299925 - OEIS

%I #6 Feb 23 2018 11:12:06

%S 1,1,2,2,4,6,8,4,12,16,16,16,32,40,44,8,64,44,128,52,136,96,256,40,88,

%T 224,88,152,512,204,1024,16,384,512,360,136,2048,1152,1024,152,4096,

%U 744,8192,416,496,2560,16384,96,720,496,2624,1088,32768,360,1216,504

%N Number of chains in Young's lattice from () to the partition with Heinz number n.

%C a(n) is the number of normal generalized Young tableaux, of shape the integer partition with Heinz number n, with all rows and columns weakly increasing and all regions skew-partitions. A generalized Young tableau of shape y is an array obtained by replacing the dots in the Ferrers diagram of y with positive integers. A tableau is normal if its entries span an initial interval of positive integers. The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

%e The a(9) = 12 tableaux:

%e 1 3   1 2

%e 2 4   3 4

%e .

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

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

%e .

%e 1 2   1 2   1 1   1 1

%e 2 2   1 2   2 2   1 2

%e .

%e 1 1

%e 1 1

%e The a(9) = 12 chains of Heinz numbers:

%e 1<9,

%e 1<2<9, 1<3<9, 1<4<9, 1<6<9,

%e 1<2<3<9, 1<2<4<9, 1<2<6<9, 1<3<6<9, 1<4<6<9,

%e 1<2<3<6<9, 1<2<4<6<9.

%t primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];

%t hncQ[a_,b_]:=And@@GreaterEqual@@@Transpose[PadRight[{Reverse[primeMS[b]],Reverse[primeMS[a]]}]];

%t chns[x_,y_]:=chns[x,y]=Join[{{x,y}},Join@@Function[c,Append[#,y]&/@chns[x,c]]/@Select[Range[x+1,y-1],hncQ[x,#]&&hncQ[#,y]&]];

%t Table[Length[chns[1,n]],{n,30}]

%Y Cf. A000085, A001222, A056239, A063834, A112798, A122111, A138178, A153452, A238690, A296150, A296188, A296561, A297388, A299202.

%K nonn

%O 1,3

%A _Gus Wiseman_, Feb 21 2018