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## co.combinatorics – Combinatorial/diagrammatic fashions totally free significance partition polynomials

For those that already know free chance idea and random matrices is my query

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

Associated partition and decreased polynomials have bushes, 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 by way of 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 newer publication cited beneath, 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**:

The formal basic moments $ has {m} _n $ and accumulators $ has {c} _n $ are by way of an inverse pair of capabilities – 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 functions in absolute evaluation and mathematical physics.

An equal relationship can breathe developed for the free moments $ m_n $ and free accumulators $ c_n $. The Voiculescu polynomials / free cumulative partition polynomials of A134264 generate the free moments from the free cumulants, which might 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 referred to 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**:

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 bushes, and many others.,

$ 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 decreased 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 capabilities 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 decreased polynomials obtained by setting. can breathe obtained $ m_n = -t $ and eradicating the next total 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 beneath.

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

To get the distribution polynomials with decreased 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 will probably breathe acknowledged 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 decreased polynomials and in linked arrays, resembling A055151, pose refined fashions for the complete partition polynomial, e.g. Schroeder paths and bushes, in addition to connections to varied algebras.

I’ve drawn the 2 completely different Schroeder path representations introduced 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 establishing combinatorial / diagrammatic fashions,

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

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