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 A299925 Number of chains in Young's lattice from () to the partition with Heinz number n. 16
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 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). LINKS Table of n, a(n) for n=1..56. EXAMPLE 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. 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}] CROSSREFS Cf. A000085, A001222, A056239, A063834, A112798, A122111, A138178, A153452, A238690, A296150, A296188, A296561, A297388, A299202. Sequence in context: A089284 A297106 A357877 * A350652 A081488 A199117 Adjacent sequences: A299922 A299923 A299924 * A299926 A299927 A299928 KEYWORD nonn AUTHOR Gus Wiseman, Feb 21 2018 STATUS approved

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Last modified August 6 00:15 EDT 2024. Contains 374957 sequences. (Running on oeis4.)