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n = 3 A = AlternatingSignMatrices(2*n) A len(A) B = [] for M in A: if M.rotate_ccw().rotate_ccw() == M: B.append(M) len(B) C = [] for M in B: flag = true for j in range(1,n+1): for k in range(1,n+1): if sum(sum(M.to_matrix().submatrix(n-k,0,2*k,j))) > k: flag = false if sum(sum(M.to_matrix().submatrix(0,n-j,k,2*j))) > j: flag = false if flag: C.append(M) #else: #print M #print len(C)
Alternating sign matrices of size 6 7436 140 132
n=3 W = WeylGroup(["B",n]) P = W.bruhat_poset(facade = True) P P.show() #len(P.cuts()) #P.completion_by_cuts() #for n >= 4, not all bi-grassmannians are join-irreducible so this won't work my_list = [] for p in P: if len(p.descents())==1 and len(p.inverse().descents()) == 1: my_list.append(p) Q = P.subposet(my_list) Q Q.show() Q.order_ideals_lattice()
Finite poset containing 48 elements
Finite poset containing 19 elements
Finite lattice containing 132 elements
n=3 W = WeylGroup(["A",n]) P = W.bruhat_poset(facade = True) P len(P.cuts()) my_list = [] for p in P: if len(p.descents())==1 and len(p.inverse().descents()) == 1: my_list.append(p) Q = P.subposet(my_list) Q Q.show() Q.order_ideals_lattice()
Finite poset containing 24 elements 42 Finite poset containing 10 elements
Finite lattice containing 42 elements
n=3 W = WeylGroup(["A",n]) P = W.bruhat_poset(facade = True) P.show(figsize=10)
Q = P.completion_by_cuts() Q Q.show(figsize=20,label_elements=false)
Finite lattice containing 42 elements
A = AlternatingSignMatrices(4) A len(A) for M in A: show(M)
Alternating sign matrices of size 4 42
(1000010000100001)\displaystyle \left(\begin{array}{rrrr} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right)
(0100100000100001)\displaystyle \left(\begin{array}{rrrr} 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right)
(1000001001000001)\displaystyle \left(\begin{array}{rrrr} 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right)
(0100111001000001)\displaystyle \left(\begin{array}{rrrr} 0 & 1 & 0 & 0 \\ 1 & -1 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right)
(0010100001000001)\displaystyle \left(\begin{array}{rrrr} 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right)
(0100001010000001)\displaystyle \left(\begin{array}{rrrr} 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right)
(0010010010000001)\displaystyle \left(\begin{array}{rrrr} 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right)
(1000010000010010)\displaystyle \left(\begin{array}{rrrr} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \end{array}\right)
(0100100000010010)\displaystyle \left(\begin{array}{rrrr} 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \end{array}\right)
(1000001001110010)\displaystyle \left(\begin{array}{rrrr} 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & -1 & 1 \\ 0 & 0 & 1 & 0 \end{array}\right)
(0100111001110010)\displaystyle \left(\begin{array}{rrrr} 0 & 1 & 0 & 0 \\ 1 & -1 & 1 & 0 \\ 0 & 1 & -1 & 1 \\ 0 & 0 & 1 & 0 \end{array}\right)
(0010100001110010)\displaystyle \left(\begin{array}{rrrr} 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & -1 & 1 \\ 0 & 0 & 1 & 0 \end{array}\right)
(0100001010110010)\displaystyle \left(\begin{array}{rrrr} 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 1 & 0 & -1 & 1 \\ 0 & 0 & 1 & 0 \end{array}\right)
(0010010010110010)\displaystyle \left(\begin{array}{rrrr} 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & -1 & 1 \\ 0 & 0 & 1 & 0 \end{array}\right)
(1000000101000010)\displaystyle \left(\begin{array}{rrrr} 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{array}\right)
(0100110101000010)\displaystyle \left(\begin{array}{rrrr} 0 & 1 & 0 & 0 \\ 1 & -1 & 0 & 1 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{array}\right)
(0010101101000010)\displaystyle \left(\begin{array}{rrrr} 0 & 0 & 1 & 0 \\ 1 & 0 & -1 & 1 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{array}\right)
(0001100001000010)\displaystyle \left(\begin{array}{rrrr} 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{array}\right)
(0100000110000010)\displaystyle \left(\begin{array}{rrrr} 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{array}\right)
(0010011110000010)\displaystyle \left(\begin{array}{rrrr} 0 & 0 & 1 & 0 \\ 0 & 1 & -1 & 1 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{array}\right)
(0001010010000010)\displaystyle \left(\begin{array}{rrrr} 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{array}\right)
(1000001000010100)\displaystyle \left(\begin{array}{rrrr} 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 \end{array}\right)
(0100111000010100)\displaystyle \left(\begin{array}{rrrr} 0 & 1 & 0 & 0 \\ 1 & -1 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 \end{array}\right)
(0010100000010100)\displaystyle \left(\begin{array}{rrrr} 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 \end{array}\right)
(0100001011010100)\displaystyle \left(\begin{array}{rrrr} 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 1 & -1 & 0 & 1 \\ 0 & 1 & 0 & 0 \end{array}\right)
(0010010011010100)\displaystyle \left(\begin{array}{rrrr} 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & -1 & 0 & 1 \\ 0 & 1 & 0 & 0 \end{array}\right)
(1000000100100100)\displaystyle \left(\begin{array}{rrrr} 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \end{array}\right)
(0100110100100100)\displaystyle \left(\begin{array}{rrrr} 0 & 1 & 0 & 0 \\ 1 & -1 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \end{array}\right)
(0010101100100100)\displaystyle \left(\begin{array}{rrrr} 0 & 0 & 1 & 0 \\ 1 & 0 & -1 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \end{array}\right)
(0001100000100100)\displaystyle \left(\begin{array}{rrrr} 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \end{array}\right)
(0100000111100100)\displaystyle \left(\begin{array}{rrrr} 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & -1 & 1 & 0 \\ 0 & 1 & 0 & 0 \end{array}\right)
(0010011111100100)\displaystyle \left(\begin{array}{rrrr} 0 & 0 & 1 & 0 \\ 0 & 1 & -1 & 1 \\ 1 & -1 & 1 & 0 \\ 0 & 1 & 0 & 0 \end{array}\right)
(0001010011100100)\displaystyle \left(\begin{array}{rrrr} 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 \\ 1 & -1 & 1 & 0 \\ 0 & 1 & 0 & 0 \end{array}\right)
(0010000110000100)\displaystyle \left(\begin{array}{rrrr} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \end{array}\right)
(0001001010000100)\displaystyle \left(\begin{array}{rrrr} 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \end{array}\right)
(0100001000011000)\displaystyle \left(\begin{array}{rrrr} 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \end{array}\right)
(0010010000011000)\displaystyle \left(\begin{array}{rrrr} 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \end{array}\right)
(0100000100101000)\displaystyle \left(\begin{array}{rrrr} 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0 \end{array}\right)
(0010011100101000)\displaystyle \left(\begin{array}{rrrr} 0 & 0 & 1 & 0 \\ 0 & 1 & -1 & 1 \\ 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0 \end{array}\right)
(0001010000101000)\displaystyle \left(\begin{array}{rrrr} 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0 \end{array}\right)
(0010000101001000)\displaystyle \left(\begin{array}{rrrr} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \end{array}\right)
(0001001001001000)\displaystyle \left(\begin{array}{rrrr} 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \end{array}\right)
(0100001000011000)\displaystyle \left(\begin{array}{rrrr} 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \end{array}\right)
(0010010000011000)\displaystyle \left(\begin{array}{rrrr} 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \end{array}\right)
(0100000100101000)\displaystyle \left(\begin{array}{rrrr} 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0 \end{array}\right)
(0010011100101000)\displaystyle \left(\begin{array}{rrrr} 0 & 0 & 1 & 0 \\ 0 & 1 & -1 & 1 \\ 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0 \end{array}\right)
(0001010000101000)\displaystyle \left(\begin{array}{rrrr} 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0 \end{array}\right)
(0010000101001000)\displaystyle \left(\begin{array}{rrrr} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \end{array}\right)
(0001001001001000)\displaystyle \left(\begin{array}{rrrr} 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \end{array}\right)
42
(1000010000100001)\displaystyle \left(\begin{array}{rrrr} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right)
(0100100000100001)\displaystyle \left(\begin{array}{rrrr} 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right)
(1000001001000001)\displaystyle \left(\begin{array}{rrrr} 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right)
(0100111001000001)\displaystyle \left(\begin{array}{rrrr} 0 & 1 & 0 & 0 \\ 1 & -1 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right)
(0010100001000001)\displaystyle \left(\begin{array}{rrrr} 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right)
(0100001010000001)\displaystyle \left(\begin{array}{rrrr} 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right)
(0010010010000001)\displaystyle \left(\begin{array}{rrrr} 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right)
(1000010000010010)\displaystyle \left(\begin{array}{rrrr} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \end{array}\right)
(0100100000010010)\displaystyle \left(\begin{array}{rrrr} 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \end{array}\right)
(1000001001110010)\displaystyle \left(\begin{array}{rrrr} 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & -1 & 1 \\ 0 & 0 & 1 & 0 \end{array}\right)
(0100111001110010)\displaystyle \left(\begin{array}{rrrr} 0 & 1 & 0 & 0 \\ 1 & -1 & 1 & 0 \\ 0 & 1 & -1 & 1 \\ 0 & 0 & 1 & 0 \end{array}\right)
(0010100001110010)\displaystyle \left(\begin{array}{rrrr} 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & -1 & 1 \\ 0 & 0 & 1 & 0 \end{array}\right)
(0100001010110010)\displaystyle \left(\begin{array}{rrrr} 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 1 & 0 & -1 & 1 \\ 0 & 0 & 1 & 0 \end{array}\right)
(0010010010110010)\displaystyle \left(\begin{array}{rrrr} 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & -1 & 1 \\ 0 & 0 & 1 & 0 \end{array}\right)
(1000000101000010)\displaystyle \left(\begin{array}{rrrr} 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{array}\right)
(0100110101000010)\displaystyle \left(\begin{array}{rrrr} 0 & 1 & 0 & 0 \\ 1 & -1 & 0 & 1 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{array}\right)
(0010101101000010)\displaystyle \left(\begin{array}{rrrr} 0 & 0 & 1 & 0 \\ 1 & 0 & -1 & 1 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{array}\right)
(0001100001000010)\displaystyle \left(\begin{array}{rrrr} 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{array}\right)
(0100000110000010)\displaystyle \left(\begin{array}{rrrr} 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{array}\right)
(0010011110000010)\displaystyle \left(\begin{array}{rrrr} 0 & 0 & 1 & 0 \\ 0 & 1 & -1 & 1 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{array}\right)
(0001010010000010)\displaystyle \left(\begin{array}{rrrr} 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{array}\right)
(1000001000010100)\displaystyle \left(\begin{array}{rrrr} 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 \end{array}\right)
(0100111000010100)\displaystyle \left(\begin{array}{rrrr} 0 & 1 & 0 & 0 \\ 1 & -1 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 \end{array}\right)
(0010100000010100)\displaystyle \left(\begin{array}{rrrr} 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 \end{array}\right)
(0100001011010100)\displaystyle \left(\begin{array}{rrrr} 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 1 & -1 & 0 & 1 \\ 0 & 1 & 0 & 0 \end{array}\right)
(0010010011010100)\displaystyle \left(\begin{array}{rrrr} 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & -1 & 0 & 1 \\ 0 & 1 & 0 & 0 \end{array}\right)
(1000000100100100)\displaystyle \left(\begin{array}{rrrr} 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \end{array}\right)
(0100110100100100)\displaystyle \left(\begin{array}{rrrr} 0 & 1 & 0 & 0 \\ 1 & -1 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \end{array}\right)
(0010101100100100)\displaystyle \left(\begin{array}{rrrr} 0 & 0 & 1 & 0 \\ 1 & 0 & -1 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \end{array}\right)
(0001100000100100)\displaystyle \left(\begin{array}{rrrr} 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \end{array}\right)
(0100000111100100)\displaystyle \left(\begin{array}{rrrr} 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & -1 & 1 & 0 \\ 0 & 1 & 0 & 0 \end{array}\right)
(0010011111100100)\displaystyle \left(\begin{array}{rrrr} 0 & 0 & 1 & 0 \\ 0 & 1 & -1 & 1 \\ 1 & -1 & 1 & 0 \\ 0 & 1 & 0 & 0 \end{array}\right)
(0001010011100100)\displaystyle \left(\begin{array}{rrrr} 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 \\ 1 & -1 & 1 & 0 \\ 0 & 1 & 0 & 0 \end{array}\right)
(0010000110000100)\displaystyle \left(\begin{array}{rrrr} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \end{array}\right)
(0001001010000100)\displaystyle \left(\begin{array}{rrrr} 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \end{array}\right)
n=3 W = WeylGroup(["B",n]) P = W.bruhat_poset(facade = True) P P.show()
Finite poset containing 48 elements
W = WeylGroup(["B",4]) print W.simple_reflection(1) print print W.simple_reflection(2) print print W.simple_reflection(3) print print W.simple_reflection(4) print
[0 1 0 0] [1 0 0 0] [0 0 1 0] [0 0 0 1] [1 0 0 0] [0 0 1 0] [0 1 0 0] [0 0 0 1] [1 0 0 0] [0 1 0 0] [0 0 0 1] [0 0 1 0] [ 1 0 0 0] [ 0 1 0 0] [ 0 0 1 0] [ 0 0 0 -1]