# Obtain all integer partitions for a given integer

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Farid Salazar Wong on 1 Jul 2015
Edited: Bruno Luong on 19 Sep 2020
Is there a way to compute all integer partitions for a given integer?
For example n=4
we have
4+ 0 + 0 +0
3+ 1 + 0 +0
2+ 2 + 0 +0
2+ 1 + 1 +0
1+ 1 + 1 +1
I would like to obtain a matrix
[4, 0, 0, 0;
3, 1, 0, 0;
2, 2, 0, 0;
2, 1, 1, 0;
1, 1, 1, 1]

Matt J on 1 Jul 2015
This seems to do it. It uses NSUMK ( Download ). For example,
[~,x] = nsumk(4,4);
result=flipud(unique(sort(x,2,'descend'),'rows'))
result =
4 0 0 0
3 1 0 0
2 2 0 0
2 1 1 0
1 1 1 1
Walter Roberson on 19 Sep 2020
Line 11 of nsumk.m is a comment, so that does not make any sense
% is a natural way to list coefficients of polynomials in N variables of degree K.
unless you also provided your own nsumk.m ?
Also, which MATLAB version are you using?

Bruno Luong on 19 Sep 2020
Edited: Bruno Luong on 19 Sep 2020
I wrote a short function that doesn't need to post-proceesed with UNIQUE (wast of time and memory) when using with NSUMK or allvL1 (order matter)
function v = Partition(n, lgt)
% v = Partition(n)
% INPUT
% n: non negative integer
% lgt: optional non negative integer
% OUTPUT:
% v: (m x lgt) non-negative integer array such as sum(v,2)==n
% each row of v is descending sorted
% v contains all possible combinations
% m = P(n) in case lgt == n, where P is the partition function
% v is (dictionnary) sorted
% Algorithm:
% Recursive
% Example:
% >> Partition(5)
%
% ans =
%
% 5 0 0 0 0
% 4 1 0 0 0
% 3 2 0 0 0
% 3 1 1 0 0
% 2 2 1 0 0
% 2 1 1 1 0
% 1 1 1 1 1
if nargin < 2
lgt = n;
end
v = PartR(lgt+1, n, Inf);
end % Partition
%% Recursive engine of integer partition
function v = PartR(n, L1, head)
if rcall
end
if n <= 2
if ~rcall
v = L1;
else
v = zeros(0, n, class(L1));
end
else % recursive call
v = arrayfun(@(j) PartR(n-1, L1, j), j(:), 'UniformOutput', false);
v = cat(1,v{:});
if rcall
end
end
end % PartR