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A304710
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Number of partitions of 2n whose Ferrers-Young diagram cannot be tiled with dominoes.
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4
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0, 0, 0, 1, 2, 6, 12, 25, 46, 85, 146, 250, 410, 666, 1053, 1648, 2527, 3840, 5747, 8525, 12496, 18172, 26165, 37408, 53038, 74714, 104502, 145315, 200808, 276030, 377339, 513342, 694925, 936590, 1256670, 1679310, 2234994, 2963430, 3914701, 5153434, 6760937
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OFFSET
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0,5
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REFERENCES
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Also the number of partitions of 2n where the number of odd parts in even positions differs from the number of odd parts in odd positions.
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LINKS
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Alois P. Heinz, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Ferrers Diagram
Wikipedia, Domino
Wikipedia, Domino tiling
Wikipedia, Ferrers diagram
Wikipedia, Mutilated chessboard problem
Wikipedia, Partition (number theory)
Wikipedia, Young tableau, Diagrams
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FORMULA
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a(n) = A058696(n) - A000712(n) = A000041(2*n) - A000712(n).
a(n) = A144064(2*n,1) - A144064(n,2).
a(n) ~ exp(2*Pi*sqrt(n/3)) / (8*sqrt(3)*n) * (1 - 2/(3^(1/4)*n^(1/4)) - (sqrt(3)/(2*Pi) + Pi/(48*sqrt(3))) / sqrt(n) + (Pi/(6*3^(3/4)) + 15*3^(1/4)/(8*Pi)) / n^(3/4)). - Vaclav Kotesovec, May 25 2018
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EXAMPLE
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a(3) = 1: the Ferrers-Young diagram of 321 cannot be tiled with dominoes because the numbers of white and black squares (when colored like a chessboard) are different but each domino covers exactly one white and one black square:
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a(4) = 2: 32111, 521.
a(5) = 6: 3211111, 32221, 4321, 52111, 541, 721.
a(6) = 12: 321111111, 3222111, 33321, 432111, 5211111, 52221, 54111, 543, 6321, 72111, 741, 921.
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MAPLE
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b:= proc(n, i, p, c) option remember; `if`(n=0, `if`(c=0, 0, 1), `if`(i<1, 0,
b(n, i-1, p, c)+b(n-i, min(n-i, i), -p, c+`if`(i::odd, p, 0))))
end:
a:= n-> b(2*n$2, 1, 0):
seq(a(n), n=0..50);
# second Maple program:
a:= n-> (p-> p(2*n)-add(p(j)*p(n-j), j=0..n))(combinat[numbpart]):
seq(a(n), n=0..50);
# third Maple program:
b:= proc(n, k) option remember; `if`(n=0, 1, add(
numtheory[sigma](j)*b(n-j, k), j=1..n)*k/n)
end:
a:= n-> b(2*n, 1)-b(n, 2):
seq(a(n), n=0..50);
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CROSSREFS
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Column k=0 of A304789.
Cf. A000041, A000712, A058696, A144064, A182616, A304662.
Sequence in context: A294565 A210068 A210633 * A137829 A262196 A261667
Adjacent sequences: A304707 A304708 A304709 * A304711 A304712 A304713
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KEYWORD
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nonn
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AUTHOR
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Alois P. Heinz, May 17 2018
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STATUS
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approved
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