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 A297388 Number of pairs (p,q) of partitions such that q is a partition of n and p <= q (diagram containment). 18
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS 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). REFERENCES R. Stanley, Enumerative Combinatorics, vol. 1, second ed., Cambridge Univ. Press, 2012. LINKS Alois P. Heinz, Table of n, a(n) for n = 0..1000 FORMULA a(n) = A000041(n) + Sum_{k=1..n} A259478(n,k). - Alois P. Heinz, Jan 10 2018 EXAMPLE For n = 2 the six pairs are (empty set,2), (1,2), (2,2), (empty set,11), (1,11), (11,11). MAPLE 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 MATHEMATICA 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 *) CROSSREFS Cf. A000041, A259478, A305023. Sequence in context: A075632 A288979 A289048 * A115217 A094687 A219753 Adjacent sequences:  A297385 A297386 A297387 * A297389 A297390 A297391 KEYWORD nonn,easy AUTHOR Richard Stanley, Dec 29 2017 STATUS approved

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Last modified January 22 07:39 EST 2020. Contains 331139 sequences. (Running on oeis4.)