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Monotone Triangle


A monotone triangle (also called a strict Gelfand pattern or a gog triangle) of order n is a number triangle with n numbers along each side and the base containing entries between 1 and n 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 n and alternating sign matrices of order n obtained by letting the kth row of the triangle equal the positions of 1s in the sum of the first k 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
(1)
(0,0,0,1,0)->4
(2)
(0,0,0,1,0)+(0,1,0,-1,1)=(0,1,0,0,1)->2 5
(3)
(0,1,0,0,1)+(1,-1,0,1,0)=(1,0,0,1,1)->1 4 5
(4)
(1,0,0,1,1)+(0,0,1,0,0)=(1,0,1,1,1)->1 3 4 5
(5)
(1,0,1,1,1)+(0,1,0,0,0)=(1,1,1,1,1)->1 2 3 4 5
(6)

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References

Bressoud, D. and Propp, J. "How the Alternating Sign Matrix Conjecture was Solved." Not. Amer. Math. Soc. 46, 637-646.

Referenced on Wolfram|Alpha

Monotone Triangle

Cite this as:

Weisstein, Eric W. "Monotone Triangle." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/MonotoneTriangle.html

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