

A115994


Triangle read by rows: T(n,k) is number of partitions of n with Durfee square of size k (n>=1; 1<=k<=floor(sqrt(n))).


83



1, 2, 3, 4, 1, 5, 2, 6, 5, 7, 8, 8, 14, 9, 20, 1, 10, 30, 2, 11, 40, 5, 12, 55, 10, 13, 70, 18, 14, 91, 30, 15, 112, 49, 16, 140, 74, 1, 17, 168, 110, 2, 18, 204, 158, 5, 19, 240, 221, 10, 20, 285, 302, 20, 21, 330, 407, 34, 22, 385, 536, 59, 23, 440, 698, 94, 24, 506, 896, 149, 25
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OFFSET

1,2


COMMENTS

Row n has floor(sqrt(n)) terms. Row sums yield A000041. Column 2 yields A006918. sum(k*T(n,k),k=1..floor(sqrt(n)))=A115995.
T(n,k) is number of partitions of nk^2 into parts of 2 kinds with at most k of each kind.
The limit of the diagonals is A000712 (partitions into parts of two kinds). In particular, if 0<=m<=n, T(n(n+1)/2 + m, n) = A000712(m). These partitions in this range can be viewed as an equilateral right triangle of side n, with one partition appended on the top (at the left) and another appended on the right.  Franklin T. AdamsWatters, Jan 11 2006


REFERENCES

G. E. Andrews, The Theory of Partitions, AddisonWesley, 1976 (pp. 2728).
G. E. Andrews and K. Eriksson, Integer Partitions, Cambridge Univ. Press, 2004 (pp. 7578).


LINKS

E. R. Canfield, S. Corteel, C. D. Savage, Durfee Polynomials, Electronic Journal of Combinatorics 5 (1998), #R32.


FORMULA

G.f.: sum(k>=1, t^k*q^(k^2)/product(j=1..k, (1q^j)^2 ) ).
T(n,k) = Sum_{i=0}^{nk^2} P*(i,k)*P*(nk^2i), where P*(n,k) = P(n+k,k) is the number of partitions of n objects into at most k parts.


EXAMPLE

T(5,2) = 2 because the only partitions of 5 having Durfee square of size 2 are [3,2] and [2,2,1]; the other five partitions ([5], [4,1], [3,1,1], [2,1,1,1] and [1,1,1,1,1]) have Durfee square of size 1.
Triangle starts:
1;
2;
3;
4, 1;
5, 2;
6, 5;
7, 8;
8, 14;
9, 20, 1;
...


MAPLE

g:=sum(t^k*q^(k^2)/product((1q^j)^2, j=1..k), k=1..40): gser:=series(g, q=0, 32): for n from 1 to 27 do P[n]:=coeff(gser, q^n) od: for n from 1 to 27 do seq(coeff(P[n], t^j), j=1..floor(sqrt(n))) od; # yields sequence in triangular form
# second Maple program:
b:= proc(n, i) option remember;
`if`(n=0, 1, `if`(i<1, 0, b(n, i1)+`if`(i>n, 0, b(ni, i))))
end:
T:= (n, k)> add(b(m, k)*b(nk^2m, k), m=0..nk^2):
seq(seq(T(n, k), k=1..floor(sqrt(n))), n=1..30); # Alois P. Heinz, Apr 09 2012


MATHEMATICA

Map[Select[#, #>0&]&, Drop[Transpose[Table[CoefficientList[ Series[x^(n^2)/Product[1x^i, {i, 1, n}]^2, {x, 0, nn}], x], {n, 1, 10}]], 1]] //Grid (* Geoffrey Critzer, Sep 27 2013 *)
b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, b[n, i1] + If[i>n, 0, b[ni, i]]]]; T[n_, k_] := Sum[b[m, k]*b[nk^2m, k], {m, 0, nk^2}]; Table[T[n, k], {n, 1, 30}, {k, 1, Sqrt[n]}] // Flatten (* JeanFrançois Alcover, Dec 25 2015, after Alois P. Heinz *)


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STATUS

approved



