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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

REFERENCES

M. S. Cheema, Tables of partitions of Gaussian integers, National Institute of Sciences of India, New Delhi, 1956.

LINKS

Alois P. Heinz, Rows n = 0..75, flattened

F. C. Auluck, On partitions of bipartite numbers, Proc. Cambridge Philos. Soc. 49, (1953). 72-83.

F. C. Auluck, On partitions of bipartite numbers, annotated scan of a few pages.

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.

Reinhard Zumkeller, Haskell programs for A201376, A054225, A201377, A054242

FORMULA

G.f.: Product_{ i=1..infinity, 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.

Columns 0-10: A000041, A000070, A000291, A000412, A000465, A000491, A002755-A002759.

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 * A228482 A091822 A173022

Adjacent sequences:  A054222 A054223 A054224 * A054226 A054227 A054228

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 26 10:42 EDT 2017. Contains 284111 sequences.