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 A321854 Irregular triangle where T(H(u),H(v)) is the number of ways to partition the Young diagram of u into vertical sections whose sizes are the parts of v, where H is Heinz number. 10
 1, 1, 0, 1, 1, 1, 0, 0, 1, 0, 2, 1, 0, 0, 0, 0, 1, 1, 3, 1, 0, 2, 0, 4, 1, 0, 0, 0, 3, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 5, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 4, 1, 0, 0, 0, 6, 0, 6, 1, 1, 3, 4, 6, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,11 COMMENTS Row n has length A000041(A056239(n)). A vertical section is a partial Young diagram with at most one square in each row. LINKS EXAMPLE Triangle begins:   1   1   0  1   1  1   0  0  1   0  2  1   0  0  0  0  1   1  3  1   0  2  0  4  1   0  0  0  3  1   0  0  0  0  0  0  1   0  2  2  5  1   0  0  0  0  0  0  0  0  0  0  1   0  0  0  0  0  4  1   0  0  0  6  0  6  1   1  3  4  6  1   0  0  0  0  0  0  0  0  0  0  0  0  0  0  1   0  0  4 10  4  8  1 The 12th row counts the following partitions of the Young diagram of (211) into vertical sections (shown as colorings by positive integers):   T(12,7) = 0: .   T(12,9) = 2:    1 2   1 2                   1     2                   2     1 .   T(12,10) = 2:   1 2   1 2                   2     1                   2     1 .   T(12,12) = 5:   1 2   1 2   1 2   1 2   1 2                   3     2     3     1     3                   3     3     2     3     1 .   T(12,16) = 1:   1 2                   3                   4 MATHEMATICA primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]]; spsu[_, {}]:={{}}; spsu[foo_, set:{i_, ___}]:=Join@@Function[s, Prepend[#, s]&/@spsu[Select[foo, Complement[#, Complement[set, s]]=={}&], Complement[set, s]]]/@Cases[foo, {i, ___}]; ptnpos[y_]:=Position[Table[1, {#}]&/@y, 1]; ptnverts[y_]:=Select[Rest[Subsets[ptnpos[y]]], UnsameQ@@First/@#&]; Table[With[{y=Reverse[primeMS[n]]}, Table[Length[Select[spsu[ptnverts[y], ptnpos[y]], Sort[Length/@#]==primeMS[k]&]], {k, Sort[Times@@Prime/@#&/@IntegerPartitions[Total[primeMS[n]]]]}]], {n, 18}] CROSSREFS Cf. A000085, A000110, A007016, A056239, A122111, A153452, A215366, A296188, A300121, A318396, A321719-A321731, A321737, A321738, A321742-A321765. Sequence in context: A143542 A072612 A116378 * A227839 A291748 A124744 Adjacent sequences:  A321851 A321852 A321853 * A321855 A321856 A321857 KEYWORD nonn,tabf AUTHOR Gus Wiseman, Nov 19 2018 STATUS approved

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Last modified September 22 13:33 EDT 2021. Contains 347607 sequences. (Running on oeis4.)