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A299925
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Number of chains in Young's lattice from () to the partition with Heinz number n.
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16
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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
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OFFSET
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1,3
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COMMENTS
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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).
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LINKS
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EXAMPLE
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The a(9) = 12 tableaux:
1 3 1 2
2 4 3 4
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1 3 1 2 1 2 1 2 1 1
2 3 3 3 2 3 1 3 2 3
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1 2 1 2 1 1 1 1
2 2 1 2 2 2 1 2
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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.
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MATHEMATICA
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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}]
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CROSSREFS
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Cf. A000085, A001222, A056239, A063834, A112798, A122111, A138178, A153452, A238690, A296150, A296188, A296561, A297388, A299202.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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