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A297388
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Number of pairs (p,q) of partitions such that q is a partition of n and p <= q (diagram containment).
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19
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1, 2, 6, 13, 30, 58, 120, 219, 413, 730, 1296, 2201, 3766, 6206, 10241, 16500, 26502, 41748, 65600, 101417, 156264, 237741, 360146, 539838, 806030, 1192365, 1756766, 2568418, 3739724, 5408247, 7791474, 11156601, 15916288, 22585112, 31933166, 44932450, 63010688
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OFFSET
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0,2
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COMMENTS
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For fixed q, the number of p is given by a determinant due to MacMahon (the case mu=empty set and n=1 of Exercise 3.149 of the reference below).
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REFERENCES
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R. Stanley, Enumerative Combinatorics, vol. 1, second ed., Cambridge Univ. Press, 2012.
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LINKS
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Alois P. Heinz, Table of n, a(n) for n = 0..1000
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FORMULA
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a(n) = A000041(n) + Sum_{k=1..n} A259478(n,k). - Alois P. Heinz, Jan 10 2018
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EXAMPLE
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For n = 2 the six pairs are (empty set,2), (1,2), (2,2), (empty set,11), (1,11), (11,11).
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MAPLE
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b:= proc(n, i, t) option remember; `if`(n=0 or i=1, 1+
`if`(t=0, 0, n), b(n, i-1, min(i-1, t))+ add(
b(n-i, min(i, n-i), min(j, n-i)), j=0..t))
end:
a:= n-> b(n$3):
seq(a(n), n=0..40); # Alois P. Heinz, Dec 29 2017
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MATHEMATICA
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b[n_, i_, t_] := b[n, i, t] = If[n == 0 || i == 1, 1 + If[t == 0, 0, n], b[n, i - 1, Min[i - 1, t]] + Sum[b[n - i, Min[i, n - i], Min[j, n - i]], {j, 0, t}]];
a[n_] := b[n, n, n];
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Apr 30 2018, after Alois P. Heinz *)
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CROSSREFS
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Cf. A000041, A259478, A305023.
Sequence in context: A075632 A288979 A289048 * A115217 A094687 A336875
Adjacent sequences: A297385 A297386 A297387 * A297389 A297390 A297391
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KEYWORD
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nonn,easy
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AUTHOR
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Richard Stanley, Dec 29 2017
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STATUS
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approved
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