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A300118 Number of skew partitions whose quotient diagram is connected and whose numerator is the integer partition with Heinz number n. 5
1, 2, 3, 3, 4, 4, 5, 4, 6, 5, 6, 5, 7, 6, 7, 5, 8, 7, 9, 6, 8, 7, 10, 6, 10, 8, 10, 7, 11, 8, 12, 6, 9, 9, 11, 8, 13, 10, 10, 7, 14, 9, 15, 8, 11, 11, 16, 7, 15, 11, 11, 9, 17, 11, 12, 8, 12, 12, 18, 9, 19, 13, 12, 7, 13, 10, 20, 10, 13, 12, 21, 9, 22, 14, 15, 11 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

The diagram of a connected skew partition is required to be connected as a polyomino but can have empty rows or columns.

LINKS

Table of n, a(n) for n=1..76.

EXAMPLE

The a(15) = 7 denominators are (), (1), (11), (22), (3), (31), (32) with diagrams:

o o o . o o . o o . . o . . . . . . o o o

o o o o . o . . o o . o o o

Missing are the two disconnected skew partitions:

. . o . . o

o o . o

MATHEMATICA

primeMS[n_]:=If[n===1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];

undcon[y_]:=Select[Tuples[Range[0, #]&/@y], Function[v, GreaterEqual@@v&&With[{r=Select[Range[Length[y]], y[[#]]=!=v[[#]]&]}, Or[Length[r]<=1, And@@Table[v[[i]]<y[[i+1]], {i, Range[Min@@r, Max@@r-1]}]]]]];

Table[Length[undcon[Reverse[primeMS[n]]]], {n, 100}]

CROSSREFS

Cf. A000085, A000898, A056239, A006958, A138178, A153452, A238690, A259479, A259480, A296150, A297388, A299925, A299926, A300056, A300060, A300120, A300122, A300123, A300124.

Sequence in context: A227861 A336751 A294991 * A256544 A321211 A336515

Adjacent sequences: A300115 A300116 A300117 * A300119 A300120 A300121

KEYWORD

nonn

AUTHOR

Gus Wiseman, Feb 25 2018

STATUS

approved

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Last modified February 7 22:58 EST 2023. Contains 360132 sequences. (Running on oeis4.)