A monotone triangle (also called a strict Gelfand pattern or a gog triangle) of order 
is a number
triangle with 
numbers along each side and the base containing entries between 1 and 
such that there is strict increase across rows and weak increase
diagonally up or down to the right. There is a bijection between monotone triangles
of order 
and alternating sign matrices of order
obtained by letting the 
th row of the triangle equal the positions of 1s in the sum
of the first 
rows of an alternating sign matrix, as
illustrated below.
![[0 0 0 1 0; 0 1 0 -1 1; 1 -1 0 1 0; 0 0 1 0 0; 0 1 0 0 0]<->4; 2 5; 1 4 5; 1 3 4 5; 1 2 3 4 5](/images/equations/MonotoneTriangle/NumberedEquation1.svg) |
(1)
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(2)
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(3)
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(4)
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(5)
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(6)
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