I am using Mag and news theme, which is a child theme of Refined Magazine. On the admin page, there seems to be no option of changing the page width. I am capable of editing php files as well, but I can’t locate the place there where I should do it. Any help is appreciated!

# Tag: refined

## co.combinatorics – Combinatorics and geometry underlying a refined Pascal matrix/Newton identities

The partition polynomials of OEIS A263633 give the coefficients of the power series/o.g.f of the multiplicative inverse (reciprocal) of a power series/o.g.f. and so give the Newton identities for transforming between complete homogeneous symmetric polynomials/functions and elementary symmetric polynomials/functions. Certain Koszul duals are related to this.

The algebraic combinatorics of the complementary reciprocal of a Taylor series/e.g.f. is governed by the antipode/refined Euler characteristic classes of the permutahedra or, equivalently, by surjective mappings, so I have an indirect geometric combinatorial interpretation of ‘scaled’ versions of the Newton identities, but I’m looking for more direct interpretations.

What combinatoric/geometric structures are enumerated by the integer coefficients of these partition polynomials for conversion of an o.g.f. into a reciprocal o.g.f.?

## co.combinatorics – Refined Inverse Plane Partition Generation Function

I have a simple question about the generation function for inverse plane partitions:

$$ sum _ { pi in RPP ( lambda)} z ^ {| pi |} = prod_ {s in lambda} frac {1} {1-z ^ {h _ { lambda} (s)}} $$

There is a natural refinement of the right side:

$$

prod_ {s in lambda} frac {1} {1-t z_1 ^ {a _ { lambda} (s)} z_2 ^ {l _ { lambda} (s)}}

$$

Or maybe just with $ t = z_1, z_2 $.

I suspect that there should be a right side equivalent to this identity, that is, to count some "refined weight" of the inverse plane partition. Maybe along diagonals? I was wondering if there is any known generating function. I think there is a natural geometric interpretation as a Poincar polynomial in $ z_2 $.

If it helps, the right side is something like $ c _ lambda (q, t) $ of Macdonald's polynomial theory.

Thank you

## Extension of the refined topology of the subspace

Leave $ mathcal {U} = tau cup tau ^ star $, and let $ tau & # 39; $ be the only minimal topology in $ X $ containing $ mathcal {U} $. As $ tau $ Y $ tau ^ star $ they are topologies, they are closed under finite intersection; and from $ tau ^ star $ is finer than the subspace topology in $ Y $, the intersection of a set in $ tau $ with a set in $ tau ^ star $ is back in $ tau ^ star $. So $ mathcal {U} $ It is closed under a finite intersection. It turns out that

$$

tau & # 39; = { mbox {all set unions in $ mathcal {U} $} }.

$$

Consequently, each set $ W $ in $ tau & # 39; $ it can be written (not uniquely) in the form $ W = U cup V $,

where $ U in tau $ Y $ V in tau ^ star $.

Yes now $ x in X $ Y $ x in W in tau & # 39; $, to write $ W = U cup V $ as previously. Yes $ x in U $, then from $ U in tau $ Y $ Y $ it is $ tau $-dense in $ X $, $ U cap Y neq varnothing $; Yes $ x in V $, then $ V $ it is a non-empty subset of $ Y $. Taken together we see that $ W cap Y neq varnothing $, so $ Y $ it is $ tau & # 39; $-dense in $ X $.

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