

A063074


Number of partitions of 2n^2 whose Ferrersplot fits within a 2n X 2n box; number of ways to cut a 2n X 2n chessboard into two equalarea pieces along a nondescending line from lower left to upper right.


7



1, 2, 8, 58, 526, 5448, 61108, 723354, 8908546, 113093022, 1470597342, 19499227828, 262754984020, 3589093760726, 49596793134484, 692260288169282, 9747120868919060, 138298900243896166, 1975688102624819336, 28396056820503468894, 410363630540693436398
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OFFSET

0,2


COMMENTS

Also the number of subsets of {1,..,4n} containing exactly 2n elements with total sum n*(4n+1) (see also A060468 for a related sequence). This is of course the same as the number of partitions of n*(4n+1) having 2n distinct parts of length at most 4n. This number is the coefficient of t^0 q^0 in the product('(t*q^k+1/(t*q^k)','k'=1..4*n).  Roland Bacher, May 02 2002
A bijection with a dissection as above of the 2n X 2n checkerboard is given by subtracting 1,2,3,..,2n of the smallest, secondsmallest, etc. element in the subset. Example for n=2: {1,2,7,8} (yields the checkerboard partition {11,22,73,84}={0,0,4,4}), {1,3,6,8} (yields {11,32,63,84}={0,1,3,4}), {1,4,5,8} (yields {0,2,2,4}), {1,4,6,7} (yields {0,2,3,3}), {3,4,5,6} (yields {2,2,2,2}), {2,4,5,7} (yields {1,2,2,3}), {2,3,6,7} (yields {1,1,3,3}), {2,3,5,8} (yields {1,1,2,4}).
Appears to be the number of random walks of length 4n, moves +/1, starting and ending at 0 and with signed area 0 under the path. It would be nice to have a lower bound of the form a(n) > c*2^{4n}/n^d.  David_Mumford(AT)brown.edu, Jun 25 2003


LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..100
H. Prodinger, On the number of partitions of 1...n into two sets of equal cardinalities and equal sums, Canad. Math. Bull. 25(1982), pp. 238241.


FORMULA

a(n) = A029895(2n) = A067059(2n, 2n) = A107110(2n, 2n^2). a(n) seems to be close to (sqrt(75)/pi)*16^n/(20n^2+6n+1).  Henry Bottomley, May 12 2005


EXAMPLE

For a 4 X 4 board (n=2) the 8 partitions are (4,4,0,0), (4,3,1,0), (4,2,1,1), (4,2,2,0), (3,3,2,0), (3,3,1,1), (3,2,2,1), (2,2,2,2).


MAPLE

b:= proc(n, i, t) option remember;
`if`(i<t or n<t*(t+1)/2 or n>t*(2*it+1)/2, 0,
`if`(n=0, 1, b(n, i1, t) +`if`(n<i, 0, b(ni, i1, t1))))
end:
a:= n> b(n*(4*n+1), 4*n, 2*n):
seq(a(n), n=0..25); # Alois P. Heinz, Jan 18 2012


MATHEMATICA

Table[ Length@Select[ IntegerPartitions[ 2n^2 ], Length[ # ] <= 2n && First[ # ] <= 2n& ], {n, 1, 5} ] or faster: Table[ T[ 2n^2, 2n, 2n ], {n, 0, 24} ] with T[ m, a, b ] as defined in A047993.
(* second program: *)
b[n_, i_, t_] := b[n, i, t] = If[i < t  n < t (t + 1)/2  n > t (2i  t + 1)/2, 0, If[n == 0, 1, b[n, i  1, t] + If[n < i, 0, b[n  i, i  1, t  1]]]]; a[n_] := b[n (4n + 1), 4n, 2n]; Table[a[n], {n, 0, 25}] (* JeanFrançois Alcover, Aug 29 2016, after Alois P. Heinz *)


CROSSREFS

Cf. A047993, A063075.
Bisection of row n=2 of A204459.  Alois P. Heinz, Jan 18 2012
Sequence in context: A229529 A007347 A185898 * A005804 A162067 A179534
Adjacent sequences: A063071 A063072 A063073 * A063075 A063076 A063077


KEYWORD

nonn


AUTHOR

Wouter Meeussen, Aug 03 2001


EXTENSIONS

More terms from Alois P. Heinz, Jan 18 2012


STATUS

approved



