## Change page width in refined magazine/mag and news theme

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## 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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## Sprinkler Price Tyco TY4151 / TY4251

An Phat Company – Sprayer sprayers wholesaler 68 degrees C – 79 degrees C – 93 degrees C – 141 degrees C throughout the country, please send us the reference price of TY4151 / TY4251.

1. Upstream / downstream spray nozzle TY4151 / 4251
It complies with the standards of a wide range of temperature variants suitable for a wide variety of uses in areas with a moderate risk of explosion, such as: supermarkets, hotel restaurants, chemical plants. , refineries …
For use in harsh environments, the TY4151 / TY4251 can be coated with an anti-corrosive coating that extends the service life. However, customers should contact us to obtain advice on the suitability of anti-corrosive materials for each specific environment to ensure the effect of fire sprinklers. Nozzle specifications:
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+ UL, C-UL
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– Maximum working pressure: 175 psi (12.1 bar).
– Nozzle size: 3/4 inch NPT.
– Spray ratio: K = 8.0 GPM / psi½ (115.2 LPM / bar½).
– activation temperature:
+ 135 ° F (57 ° C) Orange glass tube.
+ 155 ° F (68 ° C) Red glass tube.
Yellow glass tube + 175 ° F (79 ° C).
+ 200 ° F (93 ° C) Green glass tube.
+ 286 ° F (141 ° C) blue glass tube.
Purple glass tube + 360 ° F (182 ° C).

Nozzle structure:

2. Price list of sprinkler sprinklers up / down TY4151 / TY4251 In addition, An Phat offers a large number of different spray types from Tyco manufacturers, such as:
– Standard nozzles: TY3321, TY3431, TY3331 …
– Cold storage nozzles: TY3357, TY3358, TY5339 …
– Warehouse nozzles: TY5131, TY5231, TY313, TY323 …
– ….
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