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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.

%I A054225 #132 Dec 12 2024 23:29:13
%S A054225 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,
%T A054225 15,30,47,57,57,47,30,15,22,45,77,97,109,97,77,45,22,30,67,118,162,
%U A054225 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
%N 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.
%C A054225 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.
%C A054225 Or, triangle T(n,k) of bipartite partitions of n objects, k of which are black.
%C A054225 Or, number of ways to factor p^(n-k)*q^k where p and q are distinct primes.
%C A054225 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
%C A054225 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
%C A054225 Let R be the ring of power series in two countably infinite sets of variables x_1,y_1,x_2,y_2,... that are invariant under the diagonal action (i.e, the group S of permutations of positive integers acts by w(x_i)=x_{w(i)} and w(y_i)=y_{w(i)}).  Then T(n,k) is the dimension of the (n,k)-bigraded piece of R, i.e., the bihomogeneous power series of degree n in the x-variables and k in the y-variables that are S-invariant. - _Jeremy L. Martin_, Nov 27 2024
%D A054225 M. S. Cheema, Tables of partitions of Gaussian integers, National Institute of Sciences of India, New Delhi, 1956.
%D A054225 D. E. Knuth, The Art of Computer Programming, Vol. 4A, Table A-1, page 778. - _N. J. A. Sloane_, Dec 30 2018
%H A054225 Alois P. Heinz, <a href="/A054225/b054225.txt">Rows n = 0..200, flattened</a>
%H A054225 Joerg Arndt, <a href="/A054225/a054225.txt">Multiset partitions of the multisets {1^r,2^s} for r+s=5</a>.
%H A054225 F. C. Auluck, <a href="http://dx.doi.org/10.1017/S0305004100028061">On partitions of bipartite numbers</a>, Proc. Cambridge Philos. Soc. 49, (1953). 72-83.
%H A054225 F. C. Auluck, <a href="/A002774/a002774.pdf">On partitions of bipartite numbers, annotated scan of a few pages.</a>
%H A054225 F. C. Auluck, <a href="/A002774/a002774_1.pdf">On partitions of bipartite numbers</a>, Mathematical Proceedings of the Cambridge Philosophical Society, Volume 49, Issue 01, January 1953, pp. 72-83. (full article)
%H A054225 P. A. MacMahon, <a href="http://www.jstor.org/stable/90574">Memoir on symmetric functions of the roots of systems of equations</a>, Phil. Trans. Royal Soc. London, 181 (1890), 481-536; Coll. Papers II, 32-87.
%H A054225 Reinhard Zumkeller, <a href="/A054225/a054225_1.lhs.txt">Haskell programs for A201376, A054225, A201377, A054242</a>
%F A054225 G.f.: Product_{i>=1, j=0..i} 1/(1-x^(i-j)*y^j).
%F A054225 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.
%e A054225 The second row (n=1) is 1,1 since (1,0) and (0,1) each have a single partition.
%e A054225 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).
%e A054225 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).
%e A054225 The triangle begins:
%e A054225    1;
%e A054225    1,  1;
%e A054225    2,  2,  2;
%e A054225    3,  4,  4,  3;
%e A054225    5,  7,  9,  7,   5;
%e A054225    7, 12, 16, 16,  12,  7;
%e A054225   11, 19, 29, 31,  29, 19, 11;
%e A054225   15, 30, 47, 57,  57, 47, 30, 15;
%e A054225   22, 45, 77, 97, 109, 97, 77, 45, 22;
%e A054225   ...
%e A054225 A further example: T(2,2) = 9:
%e A054225 [(2,2)],
%e A054225 [(2,1),(0,1)],
%e A054225 [(2,0),(0,2)],
%e A054225 [(2,0),(0,1),(0,1)],
%e A054225 [(1,2),(1,0)],
%e A054225 [(1,1),(1,1)],
%e A054225 [(1,1),(1,0),(0,1)],
%e A054225 [(1,0),(1,0),(0,2)],
%e A054225 [(1,0),(1,0),(0,1),(0,1)].
%p A054225 read transforms; t1 := mul( mul( 1/(1-x^(i-j)*y^j), j=0..i), i=1..11): SERIES2(t1,x,y,6);
%t A054225 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. *)
%t A054225 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_ *)
%o A054225 (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 */
%o A054225 (Haskell) see Zumkeller link.
%Y A054225 See A201376 for the same triangle formatted in a different way.
%Y A054225 Columns 0-10: A000041, A000070, A000291, A000412, A000465, A000491, A002755, A002756, A002757, A002758, A002759.
%Y A054225 Row sums: A005380. a(2n, n): A002774. a(n, [n/2]): A091437. Cf. A060244.
%Y A054225 The outer edges are T(n,0) = T(0,n) = A000041(n).
%Y A054225 A054242 gives partitions into sums of distinct pairs.
%K A054225 easy,nonn,tabl,nice,look
%O A054225 0,4
%A A054225 _Marc LeBrun_, Feb 04 2000
%E A054225 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.