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A089353 Triangle read by rows: T(n,m) = number of planar partitions of n with trace m. 6
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 (list; table; graph; refs; listen; history; text; internal format)
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

Alois P. Heinz, Rows n = 1..200, flattened

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-Fran├žois 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: A277813 A200154 A208825 * A136451 A066121 A039911

Adjacent sequences:  A089350 A089351 A089352 * A089354 A089355 A089356

KEYWORD

nonn,tabl

AUTHOR

Wouter Meeussen and Vladeta Jovovic, Dec 26 2003

EXTENSIONS

Edited by Christian G. Bower, Jan 08 2004

STATUS

approved

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Last modified October 17 04:02 EDT 2019. Contains 328106 sequences. (Running on oeis4.)