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 A323450 Number of ways to fill a Young diagram with positive integers summing to n such that all rows and columns are weakly increasing. 5
 1, 1, 3, 6, 14, 26, 56, 103, 203, 374, 702, 1262, 2306, 4078, 7242, 12628, 21988, 37756, 64682, 109606, 185082, 309958, 516932, 856221, 1412461, 2316416, 3783552 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS A generalized Young tableau of shape y is an array obtained by replacing the dots in the Ferrers diagram of y with positive integers. LINKS Table of n, a(n) for n=0..26. nLab, Young Diagram. The Unapologetic Mathematician weblog, Generalized Young Tableaux. EXAMPLE The a(4) = 14 generalized Young tableaux: 4 1 3 2 2 1 1 2 1 1 1 1 . 1 2 1 1 1 2 1 1 1 1 1 3 2 2 1 1 1 1 . 1 1 1 1 1 2 1 . 1 1 1 1 The a(5) = 26 generalized Young tableaux: 5 1 4 2 3 1 1 3 1 2 2 1 1 1 2 1 1 1 1 1 . 1 2 1 1 1 3 1 2 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 4 3 3 1 2 1 2 2 1 1 1 1 . 1 1 1 1 1 2 1 1 1 1 1 1 2 1 1 1 1 1 3 2 2 1 1 1 . 1 1 1 1 1 1 1 2 1 . 1 1 1 1 1 MATHEMATICA primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]]; facs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facs[n/d], Min@@#>=d&]], {d, Rest[Divisors[n]]}]]; ptnplane[n_]:=Union[Map[primeMS, Join@@Permutations/@facs[n], {2}]]; Table[Sum[Length[Select[ptnplane[Times@@Prime/@y], And@@(LessEqual@@@Transpose[PadRight[#]/.(0->Infinity)])&]], {y, IntegerPartitions[n]}], {n, 10}] CROSSREFS Cf. A000085, A000219, A003293, A053529, A114736, A138178, A296188, A299968. Cf. A323436, A323437, A323438, A323439, A323451. Sequence in context: A049940 A265947 A358831 * A051749 A278788 A279828 Adjacent sequences: A323447 A323448 A323449 * A323451 A323452 A323453 KEYWORD nonn,more AUTHOR Gus Wiseman, Jan 16 2019 EXTENSIONS a(16)-a(26) from Seiichi Manyama, Aug 19 2020 STATUS approved

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Last modified June 13 03:07 EDT 2024. Contains 373366 sequences. (Running on oeis4.)