A089353 Triangle read by rows: T(n,m) = number of planar partitions of n with trace m.
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
Examples
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
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
- Alois P. Heinz, Rows n = 1..200, flattened
Programs
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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 *)
Formula
G.f.: Product_{k>=1} 1/(1-q*x^k)^k (with offset n=0 in x powers).
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
Extensions
Edited by Christian G. Bower, Jan 08 2004
Comments