In the 2018 Putnam, problem A-2 can be extended to general finite lattices, asking for the value of the determinant of the "meet matrix." Are there interesting families of lattices for which the matrix is nonsingular?
Finite lattice containing 6 elements
[1, 2, 3, 4, 5]
[1 0 0 0 1]
[0 1 0 0 1]
[0 0 1 0 1]
[0 0 0 1 1]
[1 1 1 1 1]
-3
-5
[1 0 0 0 0 0 1]
[0 1 0 0 0 0 1]
[0 0 1 0 0 0 1]
[0 0 0 1 0 0 1]
[0 0 0 0 1 0 1]
[0 0 0 0 0 1 1]
[1 1 1 1 1 1 1]
[[1 0 0 0 1]
[0 1 0 0 1]
[0 0 1 0 1]
[0 0 0 1 1]
[1 1 1 1 1], -3]
[(1, 1), (2, 1), (3, 1), (4, 0), (5, 1), (6, -1), (7, 1), (8, 0), (9, 0), (10, -1), (11, 1), (12, 0), (13, 1), (14, -1), (15, -1), (16, 0), (17, 1), (18, 0), (19, 1), (20, 0), (21, -1), (22, -1), (23, 1), (24, 0), (25, 0), (26, -1), (27, 0), (28, 0), (29, 1), (30, -1), (31, 1), (32, 0), (33, -1), (34, -1), (35, -1), (36, 0), (37, 1), (38, -1), (39, -1)]
[(40, 0), (41, 1), (42, -1), (43, 1), (44, 0), (45, 0), (46, -1), (47, 1), (48, 0), (49, 0), (50, 0), (51, -1), (52, 0), (53, 1), (54, 0), (55, -1), (56, 0), (57, -1), (58, -1), (59, 1), (60, 0), (61, 1), (62, -1), (63, 0), (64, 0), (65, -1), (66, -1), (67, 1), (68, 0), (69, -1), (70, -1), (71, 1), (72, 0), (73, 1), (74, -1), (75, 0), (76, 0), (77, -1), (78, -1), (79, 1)]
[(210, -1), (2310, -1), (1470, 0)]
0
[1]
Error in lines 0-1
Traceback (most recent call last):
File "/cocalc/lib/python2.7/site-packages/smc_sagews/sage_server.py", line 1188, in execute
flags=compile_flags) in namespace, locals
File "", line 1, in <module>
File "", line 3, in poset_d
File "", line 3, in <lambda>
File "", line 3, in <lambda>
File "", line 2, in chi
File "sage/structure/category_object.pyx", line 853, in sage.structure.category_object.CategoryObject.__getattr__ (build/cythonized/sage/structure/category_object.c:7759)
return self.getattr_from_category(name)
File "sage/structure/category_object.pyx", line 868, in sage.structure.category_object.CategoryObject.getattr_from_category (build/cythonized/sage/structure/category_object.c:7922)
attr = getattr_from_other_class(self, cls, name)
File "sage/cpython/getattr.pyx", line 389, in sage.cpython.getattr.getattr_from_other_class (build/cythonized/sage/cpython/getattr.c:2469)
raise AttributeError(dummy_error_message)
AttributeError: 'FinitePoset_with_category' object has no attribute 'meet'