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## co.combinatorics – Combinatorics for the motion of Virasoro / Kac-Schwarz operators: partition polynomials of free chance principle

In the background sections under, I institute the relationships between the characterizations of the motion of Virasoro / Kac-Schwarz operators of 2D gravitational fashions made in relation to the Laurent succession by Alexandrov (behold too the Kharchev ref. In his article) and fundamental partition polynomials utilizing free chance principle (and subsequently random matrix principle).

For those that already know free chance principle, my query is

**What are some combinatorial / diagrammatic fashions for the free significance partition polynomials that give the free cumulants with respect to the free moments?**

(For those that do not, learn on.)

Associated partition and diminished polynomials have timber, lattice paths, convex polyhedra, and different diagrams, in addition to Grassmann operators, positroids, Coxeter teams, and Lie derivatives distributed over them. There is plane an affiliation with monstrous moonlight through the compositional inversion of the Laurent succession with the free cumulative partition polynomials as numerators (cf. “Modular matrix fashions“by He and Jejjala). Ebrahimi-Fard et al., in a more moderen publication cited under, give a schematic formulation each for the free significance and for the free cumulative partition polynomials within the figure of non-crossing partitions.

**Background, particulars and doubtlessly helpful associations**:

*Classic formal moments and accumulations*:

The formal classical moments are with none conception of chance $ has {m} _n $ and accumulators $ has {c} _n $ are through an inverse pair of features – the exponential and the logarithm – love (in Umbral notation with e.g. $ langle ( hat {c}.) ^ n rangle = hat {c} _n $)

$$ ln[langle e^{hat{m}. x}rangle] = langle e ^ { hat {c} .x} rangle, $$

the partition polynomials of https://oeis.org/A127671 explain the cumulants in relation to the moments, whereas these of https://oeis.org/A036040 Enter the moments within the figure of cumulants. Both have quite a few interpretations in diagrammatics and enumerating combinatorics with numerous purposes in absolute evaluation and mathematical physics.

*Free formal moments and accumulators*:

Again with out the understanding of chance, an equal algebraic relationship can breathe developed for the free moments $ m_n $ and free accumulators $ c_n $ of free chance principle. The Voiculescu polynomials / free cumulative partition polynomials of A134264 generate the free moments from the free cumulants, which may breathe clear by discovering the inverse (i.e. the compositional inverse) of a proper energy succession with respect to the coefficients of its shifted reciprocal (i.e. shifted multiplicative inverse).

More exactly, for a proper energy succession or an extraordinary generator (ogf) with a vanishing ceaseless,

$$ O (x) = x + a_1 x ^ 2 + a_2 x ^ 3 + a_3x ^ 4 + cdots, $$

the related formal free cumulations are outlined by

$$ C (x) = frac {x} {O (x)} = frac {1} {1 + a_1 x + a_2 x ^ 2 + a_3x ^ 3 + cdots} = 1 + c_1 x + c_2 x ^ 2 + c_3x ^ 3 + cdots, $$

and the ogf of the formal free moments are outlined by

$$ M (x) = O ^ {(- 1)} (x) = x + m_1 x ^ 2 + m_2 x ^ 3 + m_3x ^ 4 + cdots $$

$$ = ( frac {x} {C (x)}) ^ {(- 1)} = ( frac {x} {1 + c_1 x + c_2 x ^ 2 + c_3x ^ 3 + cdots}) ^ {(-1)} $$

$$ = x + c_1 ; x ^ 2 + (c_2 + c_1 ^ 2) ; x ^ 3 + (c_3 + 3 ; c_2c_1 + c_1 ^ 3) ; x ^ 4 $$

$$ + ; (c_4 + 2 ; c_2 ^ 2 + 4 ; c_3 ; c_1 + 6 ; c_2 ; c_1 ^ 2 + c_1 ^ 4) ; z ^ 5 + cdots $$.

Vice versa,

$$ C (x) = frac {x} {O (x)} = 1 + c_1 x + c_2 x ^ 2 + c_3x ^ 3 + cdots $$

$$ = frac {x} {M ^ {(- 1)} (x)} = frac {x} {(x + m_1 x ^ 2 + m_2 x ^ 3 + m_3x ^ 4 + cdots) ^ { (-1)}} $$

$$ = frac {1} {1-m_1x + (2m_1 ^ 2-m_2) x ^ 2 + (- 5m_1 ^ 3 + 5m_1m_2-m_3) x ^ 3 + (14 m_1 ^ 4 – 21 m_1 ^ 2 m_2 + 6 m_1 m_3 + 3 m_2 ^ 2 – m_4) x ^ 4 + cdots} $$

$$ = 1 + m_1x + (- m_1 ^ 2 + m_2) x ^ 2 + (2m_1 ^ 3 -3m_2m_1 + m_3) x ^ 3 + (-5m_1 ^ 4 + 10m_2m_1 ^ 2-4m_3m_1-2m_2 ^ 2 + m_4) x ^ 4 + (14 _m1 ^ 5 – 35 m_1 ^ 3 m_2 + 15 m_1 ^ 2 m_3 + 15 m_1 m_2 ^ 2 – 5 m_1 m_4 – 5 m_2 m_3 + m_5)) x ^ 5 + cdots $$

the place the invert of $ M (x) $ expressed by the renormalized partition polynomials of A133437 (too often known as A111785) for the formal compositional inversion of a proper energy succession, the refined Euler-characteristic partition polynomials (or signed refined floor partition polynomials) of the associates.

Apart from that, the partition polynomials of A263633 for the multiplicative inversion of an influence succession can then breathe used for samples by hand:

from the OEIS entry

$$ frac {1} {1 + b_1 + b_2x + b_3x + cdots} = 1-b_1x + (b_1 ^ 2-b2) x ^ 2 + (-b_1 ^ 3 + 2b_1b_2-b_3) + cdots, $$

So

$$ m_3 = – (- m_1) ^ 3 + 2 (-m_1) (2m_1 ^ 2-m_2) – (- 5m_1 ^ 3 + 5m_1m_2 + m_3) = 2m_1 ^ 3 -3m_1m_2 + m_3. $$

*Compilation of formulation with related OEIS entries that acquire related combinatorics*:

The first moments with respect to the cumulants are given by the cumulant partition polynomials of A134264 and A125181, which enumerate non-crossing partitions (NCP), conspicuous Dyck paths, inescapable teams of timber, and so on.,

$ m_1 = c_1 = NCP_1 (1, c_1), $

$ m_2 = c_2 + c_1 ^ 2 = NCP_2 (1, c_1, c_2), $

$ m_3 = c_3 + 3 ; c_1 ; c_2 + c_1 ^ 3 = NCP_3 (1, c_1, c_2, c_3), $

$ m_4 = c_4 + 4 ; c_1 ; c_3 + 2 ; c_2 ^ 2 + 6 ; c_1 ^ 2 ; c_2 + c_1 ^ 4 = NCP_4 (1, c_1, c_2, c_3, c_4), $

with the diminished polynomials obtained by setting $ c_n = t $,

$ m_1 ^ r

$ m_2 ^ r

$ m_3 ^ r

$ m_4 ^ r

the Narayana polynomials of A001263 and A090181whose coefficients add as much as the Catalan numbers A000108,

The invert relationships are

$ c_1 = m_1 $

$ c_2 = -m_1 ^ 2 + m_2 $

$ c_3 = 2m_1 ^ 3 -3m_2m_1 + m_3 $

$ c_4 = -5m_1 ^ 4 + 10m_2m_1 ^ 2-4m_3m_1-2m_2 ^ 2 + m_4 $

$ c_5 = 14 m_1 ^ 5 + 35 m_1 ^ 3 m_2 + 15 m_1 ^ 2 m_3 + 15 m_1 m_2 ^ 2 – 5 m_1 m_4 – 5 m_2 m_3 + m_5, $

(not but within the OEIS, as I’ve already exceeded my draft restrict). These polynomials as much as $ c_4 $ can breathe create on web page 26 of “Operands of (non-crossing) partitions, interacting bialgebras and moment-cumulant relationships“by Ebrahimi-Fard, Foissy, Kock and Patras; Ex. 37 by Terry Tao’s Notes on free chance; and P. 22 of “Enumeration geometry, tau features and Heisenberg-Virasoro algebra“by Alexandrov (though he neither mentions the connection to the free moments and cumulants nor plane the free chance in any respect, he attracts connections to Virasoro / Witt group actions).

The diminished polynomials obtained by setting. can breathe obtained $ m_n = -t $ and eradicating the next general personality,

$ c_1 ^ r

$ c_2 ^ r

$ c_3 ^ r

$ c_4 ^ r

$ c_5 ^ r

these coefficients are these of A088617 and A060693 with the sizable Schröder numbers A006318 love the road totals as I display under.

The sequence of coefficients of the very best organize time period in every significance distribution polynomial is the Catalan sequence A000108, which may simply breathe proved by $ m_1 = 1 $ and clique all different moments to zero and recognize that $ x / (x + x ^ 2) ^ {(- 1)} $ provides an ogf for the Catalan sequence (in accordance with the semicircle regulation within the principle of free chance and in random matrices).

The row sums of the unsigned torque distribution polynomials are

$ 1 $

$ 1 $

$ 1 + 1 = $ 2

$ 2 + 3 + 1 = $ 6

$ 5 + 10 + 4 + 2 + 1 = $ 22

$ 14 + 35 + 15 + 15 + 5 + 5 + 1 = $ 90

Enter the preliminary phrases of the sum sequence as $ 1,2,6,22.90, these are the preliminary phrases of A006318who’ve favourited Schröder numbers.

*The related diminished polynomials*:

To get the distribution polynomials with diminished significance and to bridle the above associations, clique $ c_n = -t $, after which from the ogfs from A088617 and A086810, the shifted advocate polynomials of the associates (behold too A033282 and A126216 for the connection to the decomposition of polygons and Schroeder / Dyck lattice paths) it might probably breathe said that

$$ C (x; t) = frac {x} {O (x; t)} = frac {x} {M ^ {(- 1)} (x; t)} $$

$$ = frac {1} {1+

$$ = 1 -[; tx +(t^2+t)x^2+ (2t^3 +3t^2+t)x^3 + (5t^4+10t^3+6t^2+t)x^4+(14 t^5 + 35 t^4 + 30 t^3 + +10t^2 + t)) x^5 +cdots ;]$$

$$ = frac {1 + x + sqrt {1-2 (2t + 1) x + x ^ 2}} {2}. $$

So, combinatorial fashions described in A088617 and A060693 for the coefficients of the diminished polynomials and in linked arrays, equivalent to A055151, pose refined fashions for the complete partition polynomial, e.g. Schroeder paths and timber, in addition to connections to numerous algebras.

I’ve drawn the 2 totally different Schroeder path representations offered in A088617 and A060693 for the coefficient of the second organize time period

$ 6t ^ 2 $ in

$ c_4 ^ r

which corresponds to the 2 phrases

$ 4M_3M_1 + 2M_2 ^ $ 2 in

$ c_4 = -5m_1 ^ 4 + 10m_2m_1 ^ 2-4m_3m_1-2m_2 ^ 2 + m_4, $

and noticed that solely two of the six Schroeder paths have centerline reflection balance, however I do not behold how these paths can breathe mapped onto the partitions, therefore my query to the customers right here, love Gessel and Stanley, and their colleagues, who’re mighty more proficient than me in developing combinatorial / diagrammatic fashions,

**What are some combinatorial / diagrammatic fashions for the significance distribution polynomials?**

The (formal) Laurent succession for the Riemann illustration

$$ LC (z) = frac {C (z)} {z} = frac {1} {O (z)} = frac {1} {M ^ {(- 1)} (z)} = frac {1} {z} + c_1 + c_2z + c_3z ^ 2 + cdots $$

$$ = frac {1} {z} + m_1 + (m_2-m_1 ^ 2) z + (m_3 -3m_2m_1 + 2m_1 ^ 3) z ^ 2 + cdots, $$

has the (formal) compositional reversal

$$ LC ^ {(- 1)} (z) = LM (z) = M ( frac {1} {z}) = O ^ {(- 1)} ( frac {1} {z}) = frac {1} {z} + ; frac {m_1} {z ^ 2} + ; frac {m_2} {z ^ 3} + ; frac {m_3} {z ^ 4} + cdots $$

$$ = frac {1} {z} + ; frac {c_1} {z ^ 2} + ; frac {c_2 + c_1 ^ 2} {z ^ 3} + ; frac {c_3 + 3 ; c_2c_1 + c_1 ^ 3} {z ^ 4} + cdots. $$

Laurent succession are the everyday actors in most representations of the theories of free chance and random matrices and are mentioned specifically in He and Jejjala with account to the q-extension of Klein’s j-invariant in relation to the monster moonlight (behold p. 4 .). , 6-7, 10-11, 14-15, 20-21 and 25). Laurent succession of this kindly too toy a job in purposes within the complicated evaluation of the Faber polynomials of symmetric duty principle (behold A263916, particularly McKay and his coworkers).

Alexandrov (p. 22) has two units of Laurent succession relationships, each of that are labeled $ f (z) $ and $ tilde {f} (z) $, characterizes the actions of two subgroups of the Virasoro group – the primary, for $ FOR _ {+} $ With $ f (z) = LC ( frac {1} {z}) $ and $ tilde {f} (z) = LC ^ {(- 1)} ( frac {1} {z}) $ along with his $ b _ {- n} $, my $ c_n $; the second, for $ FOR _ {-} $ With $ f (z) = LC ^ {(- 1)} ( frac {1} {z}) $ and $ tilde {f} (z) = LC ( frac {1} {z}) $ along with his $ b_n $, my $ m_n $.

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