ASM Triangle Row

Returns a row of the alternating sign matrix triangle.

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Interface
Asm Triangle Row
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Interface

C++

Asm Triangle Row

std::vector<int>asm_triangle_row( int n )

For a given n, the value of $\inline A(j)$ represents entry $\inline A(i, j)$ of the triangular matrix, and gives the number of alternating sign matrices of order n in which the (unique) 1 in row i occurs in column j.

For example, of alternating sign matrices of order 3, there are 2 with a leading 1 in column 1:

$\left( \begin{array}{ccc} 1 & 0 & 0 \cr 0 & 1 & 0 \cr 0 & 0 & 1 \end{array} \right) \qquad \left( \begin{array}{ccc} 1 & 0 & 0 \cr 0 & 0 & 1 \cr 0 & 1 & 0 \end{array} \right)$

(2)

3 with a leading 1 in column 2:

$\left( \begin{array}{ccc} 0 & 1 & 0 \cr 1 & 0 & 0 \cr 0 & 0 & 1 \end{array} \right) \qquad \left( \begin{array}{ccc} 0 & 1 & 0 \cr 0 & 0 & 1 \cr 1 & 0 & 0 \end{array} \right) \qquad \left( \begin{array}{ccc} 0 & 1 & 0 \cr 1 & -1 & 1 \cr 0 & 1 & 0 \end{array} \right)$

(3)

and 2 with a leading 1 in column 3:

$\left( \begin{array}{ccc} 0 & 0 & 1 \cr 1 & 0 & 0 \cr 0 & 1 & 0 \end{array} \right) \qquad \left( \begin{array}{ccc} 0 & 0 & 1 \cr 0 & 1 & 0 \cr 1 & 0 & 0 \end{array} \right)$

(4)

References:

SUBSET, a C++ library of combinatorial routines, http://www.csit.fsu.edu/~burkardt/cpp_src/subset/subset.html

Example 1

#include <codecogs/maths/combinatorics/sequences/asm_triangle_row.h>
#include <iostream>
int main()
{
  std::vector<int> result = Maths::Combinatorics::Sequences::asm_triangle_row(4);
  std::cout << "Size of row: " << result.size() << std::endl;
  for (int j = 0; j < result.size(); j++)
    std::cout << result[j] << "  ";
  std::cout << std::endl;
 
  return 0;
}

Output:

Size of row: 5
42  105  135  105  42

Returns

the entries of the row

Authors

Lucian Bentea (August 2005)

Source Code

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