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A089353
Triangle read by rows: T(n,m) = number of planar partitions of n with trace m.
7
1, 2, 1, 3, 2, 1, 4, 6, 2, 1, 5, 10, 6, 2, 1, 6, 19, 14, 6, 2, 1, 7, 28, 28, 14, 6, 2, 1, 8, 44, 52, 33, 14, 6, 2, 1, 9, 60, 93, 64, 33, 14, 6, 2, 1, 10, 85, 152, 127, 70, 33, 14, 6, 2, 1, 11, 110, 242, 228, 142, 70, 33, 14, 6, 2, 1, 12, 146, 370, 404, 272, 149, 70, 33, 14, 6, 2, 1, 13
OFFSET
1,2
COMMENTS
Also number of partitions of n objects of 2 colors into k parts, each part containing at least one black object.
T(n+m, m) = A005380(n), n >= 1, for all m >= n. T(m, m) = 1 for m >= 1. See the Stanley reference Exercise 7.99. With offset n=0 a column for m=0 with the only non-vanishing entry T(0, 0) = 1 could be added. - Wolfdieter Lang, Mar 09 2015
REFERENCES
G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976 (Ch. 11, Example 5 and Ch. 12, Example 5).
R. P. Stanley, Enumerative Combinatorics, Cambridge University Press, Vol. 2, 1999; p. 365 and Exercise 7.99, p. 484 and pp. 548-549.
LINKS
FORMULA
G.f.: Product_(k>=1} 1/(1-q x^k)^k (with offset n=0 in x powers).
EXAMPLE
The triangle T(n,m) begins:
n\m 1 2 3 4 5 6 7 8 9 10 11 12 ...
1: 1
2: 2 1
3: 3 2 1
4: 4 6 2 1
5: 5 10 6 2 1
6: 6 19 14 6 2 1
7: 7 28 28 14 6 2 1
8: 8 44 52 33 14 6 2 1
9: 9 60 93 64 33 14 6 2 1
10: 10 85 152 127 70 33 14 6 2 1
11: 11 110 242 228 142 70 33 14 6 2 1
12: 12 146 370 404 272 149 70 33 14 6 2 1
... reformatted, Wolfdieter Lang, Mar 09 2015
MAPLE
b:= proc(n, i) option remember; expand(`if`(n=0, 1,
`if`(i<1, 0, add(b(n-i*j, i-1)*x^j*
binomial(i+j-1, j), j=0..n/i))))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..degree(p)))(b(n$2)):
seq(T(n), n=1..12); # Alois P. Heinz, Apr 13 2017
MATHEMATICA
b[n_, i_] := b[n, i] = Expand[If[n == 0, 1, If[i < 1, 0, Sum[b[n - i*j, i - 1]*x^j*Binomial[i + j - 1, j], {j, 0, n/i}]]]];
T[n_] := Table[Coefficient[#, x, i], {i, 1, Exponent[#, x]}]]& @ b[n, n];
Table[T[n], {n, 1, 12}] // Flatten (* Jean-François Alcover, May 19 2018, after Alois P. Heinz *)
CROSSREFS
Cf. A000219 (row sums), A005380, A005993 (trace 2), A050531 (trace 3), A089351 (trace 4).
Sequence in context: A200154 A208825 A344391 * A136451 A361894 A066121
KEYWORD
nonn,tabl
AUTHOR
EXTENSIONS
Edited by Christian G. Bower, Jan 08 2004
STATUS
approved