For a variety of sequences s, natural statistics on [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] have the same distribution as natural statistics on other equinumerous
combinatorial families, a phenomenon that was used to settle an open question about Coxeter groups in .
Notice that this last proof shows that the three sets [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (321, 231), [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII](321, 312) and[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII](321,231,312) are identical, not just equinumerous
Compare this: when it seems to us that a set could not possibly be equinumerous
with a proper subset, this seeming has no apparent correlated experience distinct from it.
To see that Cantor did not intend any restriction to definable sets, one only has to consider his remarks on the "internal determinateness" of the question whether two sets are equinumerous
This leads to the familiar proofs that the set of even numbers is equinumerous
with the set of odd numbers, and that the natural numbers are equinumerous
with the rationals.
CM,p] are equinumerous
, so it must be one-to-one as well.
But if the imagined inscriptions consist of no definite number of strokes such that we could tell what that number is, it is uncertain what an equinumerous
relation here could come to.
MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is shown to be equinumerous
to such paths with [?
Clearly the members of either set not containing an occurrence of 113 or 133, respectively, are equinumerous
by the preceding.
He can, however, be credited with the following: he derived (or showed how to derive) the whole of arithmetic in second-order logic from "Hume's principle" ("the numbers belonging to F and G are equal if and only if F is equinumerous
with G"); he derived Peano's second postulate ("every natural number has a successor") from the same principle; he succeeded, where Dedekind failed, in demonstrating the existence of an infinite system (p.
Pylyavskyy introduces and studies what he calls non-crossing tableaux, showing that they are as well equinumerous
with standard tableaux.
Alternating sign matrices (ASMs) and their equinumerous
friends, descending plane partitions (DPPs) and totally symmetric self-complementary plane partitions (TSSCPPs), have been bothering combinatorialists for decades by the lack of an explicit bijection between any two of the three sets of objects.