A299925 Number of chains in Young's lattice from () to the partition with Heinz number n.

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, 224, 88, 152, 512, 204, 1024, 16, 384, 512, 360, 136, 2048, 1152, 1024, 152, 4096, 744, 8192, 416, 496, 2560, 16384, 96, 720, 496, 2624, 1088, 32768, 360, 1216, 504

Offset: 1

Author

Gus Wiseman, Feb 21 2018

Keywords

• nonn

Comments

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).

Examples

			The a(9) = 12 tableaux:
1 3   1 2
2 4   3 4
.
1 3   1 2   1 2   1 2   1 1
2 3   3 3   2 3   1 3   2 3
.
1 2   1 2   1 1   1 1
2 2   1 2   2 2   1 2
.
1 1
1 1
The a(9) = 12 chains of Heinz numbers:
1<9,
1<2<9, 1<3<9, 1<4<9, 1<6<9,
1<2<3<9, 1<2<4<9, 1<2<6<9, 1<3<6<9, 1<4<6<9,
1<2<3<6<9, 1<2<4<6<9.
		

Crossrefs

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

Programs

Mathematica

primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
hncQ[a_,b_]:=And@@GreaterEqual@@@Transpose[PadRight[{Reverse[primeMS[b]],Reverse[primeMS[a]]}]];
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]&]];
Table[Length[chns[1,n]],{n,30}]