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A089353 Triangle read by rows: T(n,m) = number of planar partitions of n with trace m.

%I A089353 #34 Jun 04 2025 10:23:57
%S A089353 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,
%T A089353 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,
%U A089353 11,110,242,228,142,70,33,14,6,2,1,12,146,370,404,272,149,70,33,14,6,2,1,13
%N A089353 Triangle read by rows: T(n,m) = number of planar partitions of n with trace m.
%C A089353 Also number of partitions of n objects of 2 colors into k parts, each part containing at least one black object.
%D A089353 G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976 (Ch. 11, Example 5 and Ch. 12, Example 5).
%D A089353 R. P. Stanley, Enumerative Combinatorics, Cambridge University Press, Vol. 2, 1999; p. 365 and Exercise 7.99, p. 484 and pp. 548-549.
%H A089353 Alois P. Heinz, <a href="/A089353/b089353.txt">Rows n = 1..200, flattened</a>
%F A089353 G.f.: Product_{k>=1} 1/(1-q*x^k)^k (with offset n=0 in x powers).
%F A089353 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
%e A089353 The triangle T(n,m) begins:
%e A089353   n\m  1   2   3   4   5   6  7  8  9 10 11 12 ...
%e A089353   1:   1
%e A089353   2:   2   1
%e A089353   3:   3   2   1
%e A089353   4:   4   6   2   1
%e A089353   5:   5  10   6   2   1
%e A089353   6:   6  19  14   6   2   1
%e A089353   7:   7  28  28  14   6   2  1
%e A089353   8:   8  44  52  33  14   6  2  1
%e A089353   9:   9  60  93  64  33  14  6  2  1
%e A089353   10: 10  85 152 127  70  33 14  6  2  1
%e A089353   11: 11 110 242 228 142  70 33 14  6  2  1
%e A089353   12: 12 146 370 404 272 149 70 33 14  6  2  1
%e A089353   ... reformatted, _Wolfdieter Lang_, Mar 09 2015
%p A089353 b:= proc(n, i) option remember; expand(`if`(n=0, 1,
%p A089353       `if`(i<1, 0, add(b(n-i*j, i-1)*x^j*
%p A089353        binomial(i+j-1, j), j=0..n/i))))
%p A089353     end:
%p A089353 T:= n-> (p-> seq(coeff(p, x, i), i=1..degree(p)))(b(n$2)):
%p A089353 seq(T(n), n=1..12);  # _Alois P. Heinz_, Apr 13 2017
%t A089353 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 A089353 T[n_] := Table[Coefficient[#, x, i], {i, 1, Exponent[#, x]}]& @ b[n, n];
%t A089353 Table[T[n], {n, 1, 12}] // Flatten (* _Jean-François Alcover_, May 19 2018, after _Alois P. Heinz_ *)
%Y A089353 Cf. A000219 (row sums), A005380, A005993 (trace 2), A050531 (trace 3), A089351 (trace 4).
%K A089353 nonn,tabl
%O A089353 1,2
%A A089353 _Wouter Meeussen_ and _Vladeta Jovovic_, Dec 26 2003
%E A089353 Edited by _Christian G. Bower_, Jan 08 2004