The interval conjecture for pure O-sequences

Let $(1,h_1,\dotsc,h_e)$ be a sequence of natural numbers, and fix an index $i$ . Suppose that for some positive integer $\alpha$ , both

(1,h_1,\dotsc,h_i,\dotsc,h_e)

and

(1,h_1,\dotsc,h_i+\alpha,\dotsc,h_e)

are pure $O$ -sequences. Interval conjecture for pure $O$ -sequences. Then

(1,h_1,\dotsc,h_i+\beta,\dotsc,h_e)

is also a pure $O$ -sequence for every integer $\beta=1,2,\dotsc,\alpha-1$ . The conjecture was proved for socle degrees at most $3$ , but remains open in many cases and has been disproved in the four-variable case.

Equivalent formulations 1

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

The interval conjecture for pure O-sequences
Let $(1,h_1,\ldots,h_i,\ldots,h_e)$ be a pure $O$ -sequence, and let $\alpha$ be a positive integer such that $(1,h_1,\ldots,h_i+\alpha,\ldots,h_e)$ is also a pure $O$ -sequence. Interval conjecture for pure $O$ -sequences. Then $(1,h_1,\ldots,h_i+\beta,\ldots,h_e)$ is a pure $O$ -sequence for every integer $\beta=0,1,\ldots,\alpha$ . The conjecture asserts an interval property for the set of pure $O$ -sequences. The paper presents it as a major open problem and develops consequences and supporting computational evidence.
source: M. Boij, J. Migliore, R. Miro'-Roig, U. Nagel and F. Zanello, “On the shape of a pure O-sequence”, arXiv:1003.3825 (2011).

Sources & referencesView supporting material

Primary source

Huy Tài Hà, Erik Stokes and Fabrizio Zanello, “Pure O-sequences and matroid h-vectors”, arXiv:1006.0325 (2012).

combinatorial-commutative-algebra commutative-algebra pure-o-sequences

1 week ago

The interval conjecture for pure O-sequences

Equivalent formulations 1

The interval conjecture for pure O-sequences

Primary source

Progress summary

Solutions 0