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A054225 Triangle read by rows: row n (n>=0) gives the number of partitions of (n,0), (n-1,1), (n-2,2), ..., (0,n) respectively into sums of pairs. 30
1, 1, 1, 2, 2, 2, 3, 4, 4, 3, 5, 7, 9, 7, 5, 7, 12, 16, 16, 12, 7, 11, 19, 29, 31, 29, 19, 11, 15, 30, 47, 57, 57, 47, 30, 15, 22, 45, 77, 97, 109, 97, 77, 45, 22, 30, 67, 118, 162, 189, 189, 162, 118, 67, 30, 42, 97, 181, 257, 323, 339, 323, 257, 181, 97, 42, 56, 139, 267, 401, 522, 589, 589, 522, 401, 267, 139, 56 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,4
COMMENTS
By analogy with ordinary partitions (A000041). The empty partition gives T(0,0)=1 by definition. A054225 and A201377 give partitions of pairs into sums of distinct pairs. Parts (i,j) are "positive" in the sense that min {i,j} >= 0 and max {i,j} >0. The empty partition of (0,0) is counted as 1.
Or, triangle T(n,k) of bipartite partitions of n objects, k of which are black.
Or, number of ways to factor p^(n-k)*q^k where p and q are distinct primes.
In the paper by F. C. Auluck: "On partitions of bipartite numbers", p.74, in the formula for fixed m there should be factor 1/m!. The correct asymptotic formula is p(m, n) ~ (sqrt(6*n)/Pi)^m * exp(Pi*sqrt(2*n/3)) / (4*sqrt(3)*m!*n). - Vaclav Kotesovec, Feb 01 2016
T(n,k)=T(n,k-n) is the number of multiset partitions of the multiset {1^k, 2^(n-k)}, see example link. - Joerg Arndt, Jan 01 2024
REFERENCES
M. S. Cheema, Tables of partitions of Gaussian integers, National Institute of Sciences of India, New Delhi, 1956.
D. E. Knuth, The Art of Computer Programming, Vol. 4A, Table A-1, page 778. - N. J. A. Sloane, Dec 30 2018
LINKS
F. C. Auluck, On partitions of bipartite numbers, Proc. Cambridge Philos. Soc. 49, (1953). 72-83.
F. C. Auluck, On partitions of bipartite numbers, Mathematical Proceedings of the Cambridge Philosophical Society, Volume 49, Issue 01, January 1953, pp. 72-83. (full article)
P. A. MacMahon, Memoir on symmetric functions of the roots of systems of equations, Phil. Trans. Royal Soc. London, 181 (1890), 481-536; Coll. Papers II, 32-87.
FORMULA
G.f.: Product_{i>=1, j=0..i} 1/(1-x^(i-j)*y^j).
Series ends ... + 7*x^5 + 12*x^4*y + 16*x^3*y^2 + 16*x^2*y^3 + 12*x*y^4 + 7*y^5 + 5*x^4 + 7*x^3*y + 9*x^2*y^2 + 7*x*y^3 + 5*y^4 + 3*x^3 + 4*x^2*y + 4*x*y^2 + 3*y^3 + 2*x^2 + 2*x*y + 2*y^2 + x + y + 1.
EXAMPLE
The second row (n=1) is 1,1 since (1,0) and (0,1) each have a single partition.
The third row (n=2) is 2, 2, 2 from (2,0) = (1,0)+(1,0), (1,1) = (1,0)+(0,1), (0,2) = (0,1)+(0,1).
In the fourth row (n=3), T(2,1)=4 from (2,1) = (2,0)+(0,1) = (1,0)+(1,1) = (1,0)+(1,0)+(0,1).
The triangle begins:
1;
1, 1;
2, 2, 2;
3, 4, 4, 3;
5, 7, 9, 7, 5;
7, 12, 16, 16, 12, 7;
11, 19, 29, 31, 29, 19, 11;
15, 30, 47, 57, 57, 47, 30, 15;
22, 45, 77, 97, 109, 97, 77, 45, 22;
...
A further example: T(2,2) = 9:
[(2,2)],
[(2,1),(0,1)],
[(2,0),(0,2)],
[(2,0),(0,1),(0,1)],
[(1,2),(1,0)],
[(1,1),(1,1)],
[(1,1),(1,0),(0,1)],
[(1,0),(1,0),(0,2)],
[(1,0),(1,0),(0,1),(0,1)].
MAPLE
read transforms; t1 := mul( mul( 1/(1-x^(i-j)*y^j), j=0..i), i=1..11): SERIES2(t1, x, y, 6);
MATHEMATICA
rows = 11; se = Series[ Product[ 1/(1-x^(n-k)*y^k), {n, 1, rows}, {k, 0, n}], {x, 0, rows}, {y, 0, rows}]; coes = CoefficientList[ se, {x, y}]; Flatten[ Table[ coes[[n-k+1, k]], {n, 1, rows+1}, {k, 1, n}]] (* Jean-François Alcover, Nov 21 2011, after g.f. *)
p = 2; q = 3; b[n_, k_] := b[n, k] = If[n>k, 0, 1] + If[PrimeQ[n], 0, Sum[If[d>k, 0, b[n/d, d]], {d, DeleteCases[Divisors[n], 1|n]}]]; t[n_, k_] := b[p^(n-k)*q^k, p^(n-k)*q^k]; Table[t[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Mar 13 2014, after Alois P. Heinz *)
PROG
(PARI) {T(n, k) = if( n<0 || k<0, 0, polcoeff( polcoeff( prod( i=1, n, prod( j=0, i, 1 / (1 - x^i * y^j), 1 + x * O(x^n))), n), k))} /* Michael Somos, Apr 19 2005 */
(Haskell) see Zumkeller link.
CROSSREFS
See A201376 for the same triangle formatted in a different way.
Row sums: A005380. a(2n, n): A002774. a(n, [n/2]): A091437. Cf. A060244.
The outer edges are T(n,0) = T(0,n) = A000041(n).
A054242 gives partitions into sums of distinct pairs.
Sequence in context: A193921 A074829 A060243 * A322210 A228482 A091822
KEYWORD
easy,nonn,tabl,nice,look
AUTHOR
Marc LeBrun, Feb 04 2000
EXTENSIONS
Entry revised by N. J. A. Sloane, Nov 30 2011, to incorporate corrections provided by Reinhard Zumkeller, who also contributed the alternative version A201376. Once the errors were corrected, this sequence coincided with A060243, due to N. J. A. Sloane, Mar 22 2001, which included edits by Vladeta Jovovic, Mar 23 2001, and Christian G. Bower, Jan 08 2004. The two entries have now been merged.
STATUS
approved

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Last modified March 28 09:04 EDT 2024. Contains 371240 sequences. (Running on oeis4.)