A new “dinv” arising from the two part case of the shuffle conjecture
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Abstract
For a symmetric function F, the eigenoperator Δ_{ F } acts on the modified Macdonald basis of the ring of symmetric functions by \(\Delta_{F} \tilde{H}_{\mu}= F[B_{\mu}] \tilde{H}_{\mu}\). In a recent paper (Int. Math. Res. Not. 11:525–560, 2004), J. Haglund showed that the expression \(\langle\Delta_{h_{J}} E_{n,k}, e_{n}\rangle\) q,tenumerates the parking functions whose diagonal word is in the shuffle 12⋯J∪∪J+1⋯J+n with k of the cars J+1,…,J+n in the main diagonal including car J+n in the cell (1,1) by t ^{area} q ^{dinv}.
In view of some recent conjectures of Haglund–Morse–Zabrocki (Can. J. Math., doi: 10.4153/CJM20110784, 2011), it is natural to conjecture that replacing E _{ n,k } by the modified Hall–Littlewood functions \(\mathbf{C}_{p_{1}}\mathbf{C}_{p_{2}}\cdots\mathbf{C}_{p_{k}} 1\) would yield a polynomial that enumerates the same collection of parking functions but now restricted by the requirement that the Dyck path supporting the parking function touches the diagonal according to the composition p=(p _{1},p _{2},…,p _{ k }). We prove this conjecture by deriving a recursion for the polynomial \(\langle\Delta_{h_{J}} \mathbf{C}_{p_{1}}\mathbf{C}_{p_{2}}\cdots \mathbf{C}_{p_{k}} 1 , e_{n}\rangle \), using this recursion to construct a new \(\operatorname{dinv}\) statistic (which we denote \(\operatorname{ndinv}\)), then showing that this polynomial enumerates the latter parking functions by \(t^{\operatorname{area}} q^{\operatorname{ndinv}}\).
Keywords
Symmetric functions Macdonald polynomials Parking functions1 Introduction
The main result of this paper is a proof of this conjecture. Banking on the intuition gained from previous work [7] and using some of the identities developed there with the C _{ a } and B _{ b } operators we are able to derive the following basic recursion.
Theorem 1.1
2 Auxiliary identities from the Theory of Macdonald polynomials
 (1)
“power” {p _{ μ }}_{ μ },
 (2)
“monomial” {m _{ μ }}_{ μ },
 (3)
“homogeneous” {h _{ μ }}_{ μ },
 (4)
“elementary” {e _{ μ }}_{ μ },
 (5)
“forgotten” {f _{ μ }}_{ μ } and
 (6)
“Schur” {s _{ μ }}_{ μ }.
Note that the orthogonality relations in (2.10) yield us the following Macdonald polynomial expansions:
Proposition 2.1
Remark 2.2
3 Proof of the basic recursion
To establish Theorem 1.1 we need some preliminary observations. To begin we have the following reduction.
Theorem 3.1
We will give a proof of (3.1) first, and then in the following pages we will establish (3.2) after developing a few necessary identities.
Proof of (3.1)
Our next goal is to prove (3.2). To begin we have the following auxiliary identity.
Proposition 3.2
Proof
Next we have
Proposition 3.3
Proof
4 The construction of the new dinv
Before we can proceed with our construction of the new dinv, we need some preliminary observations about this family of parking functions. To begin we should note that the condition that the diagonal word be a shuffle of 12⋯J with J+1⋯J+n, together with the column increasing property of parking functions, forces the columns of the Dyck path where by the length of column i of a Dyck path D we refer to the number of NORTH steps of D of abscissa i supporting a \(\mathit{PF}\in\mathcal{PF}(J,n)\) to be of length at most 2. The reason for this is simple: as we read the cars of PF to obtain σ(PF) from right to left by diagonals starting from the highest and ending with the lowest the big cars (J+1,…,J+n) as well as the small cars (1,2,…,J) will be increasing. Thus we will never see a big car on top of a big car nor a small car on top of a small car. So the only possibility is a big car on top of a small car, i.e. columns of length most 2 as we asserted.
This yields an algorithm for constructing all the elements of the family \(\mathcal{PF}(J,n)\). Let us denote by “\(\operatorname{red}(\mathit{PF})\)”, and call it the “reduced tableau” of PF, the configuration obtained by replacing in a \(\mathit{PF}\in\mathcal{PF}(J,n)\) all big cars by a 2 and all small cars by a 1. We can simply obtain all the reduced tableaux of elements of \(\mathcal{PF}(J,n)\) by constructing first the family \(\mathcal{D}_{J,n}\) of Dyck paths of length n+J with no more than J columns of length 2 and all remaining columns of length 1. Then for each Dyck path \(D\in\mathcal{D}_{J,n}\) fill the cells adjacent to the NORTH steps of each column of length 2 by a 1 under a 2, then fill the columns of length 1 by a 1 or a 2 for a total of J ones and n twos.
Clearly each \(\mathit{PF}\in\mathcal{PF}(J,n)\) can be uniquely reconstructed from its reduced tableau by replacing all the ones by 1,2,…,J and all the twos by J+1,…,J+n by diagonals from right to left starting from the highest and ending with the lowest. It will also be clear that we need only work with reduced tableaux to construct our new dinv. However, being able to refer to the original cars will turn out to be more convenient in some of our proofs. For this reason we will work with a PF or its \(\operatorname{red}(\mathit{PF})\) interchangeably depending on the context.
This given, we have the following basic fact.
Proposition 4.1
Proof
For a while in our investigation this appeared to be a challenging puzzle. The discovery of the recursion of Theorem 1.1 completely solved this puzzle but, as we shall see, it created another puzzle.
Starting from this observation and further closer analysis of (4.8) led us to the following recursive algorithm for constructing “ndinv”.
In the case that p _{1}=1 there will be only one big car in the first section and if there are small cars they all must be on the main diagonal. In this case we can process the first section as we did for p _{1}>1. If there are no small cars then the first section consists of the single domino \(\bigl[{2\atop0}\bigr]\) and (4.8) suggests that we should simply remove it with no further ado.
In the third step we establish the equality in (4.2) by verifying the equality in the base cases.

Cut the domino sequence of \(\operatorname{red}(\mathit{PF})\) into sections starting at the dominos \(\bigl[{2\atop0}\bigr]\)
 (1)If the first section does not contain a domino \(\bigl[{1\atop0}\bigr]\)

remove its only domino \(\bigl[{2\atop0}\bigr]\) from the sequence of dominos

 (2)If the first section contains a domino \(\bigl[{1\atop0}\bigr]\), work on the first section as follows:

remove its first domino \(\bigl[{1\atop0}\bigr]\)

for each (but the first) domino \(\bigl[{2\atop a}\bigr]\) make the replacement \(\bigl[{2\atop a}\bigr] {\rightarrow}\bigl[{2\atop a1}\bigr]\)

if adjacent pairs \(\bigl[{2\atop a1}\bigr] \bigl[{1\atop a}\bigr]\) are created make the replacements \(\bigl[{2\atop a1}\bigr] \bigl[{1\atop a}\bigr] {\rightarrow}\allowbreak \bigl[{1\atop a1}\bigr] \bigl[{2\atop a}\bigr]\)

cycle the modified first section to the end of the sequence of dominos

It is clear that Φ maps the left hand side of (4.15) into the right hand side. To show that Φ is a bijection we need only show that the procedure above can be reversed to reconstruct PF from PF′ for any PF′ in the right hand side of (4.15). We will outline the salient steps of the reversed procedure.
 (1)Say \(\mathit{PF}'\in\mathcal{PF}_{J}([p_{2},\ldots, p_{k}])\) (which will only occur when p _{1}=1)

Then PF is the parking function obtained by prepending \(\bigl[{2\atop0}\bigr]\) to the domino sequence of PF′.

 (2)Say \(\mathit{PF}'\in\mathcal{PF}_{J1}([p_{2},\ldots, p_{k},1])\) (which will only occur when p _{1}=1)

Then PF is the parking function obtained by inserting \(\bigl[{1\atop0}\bigr]\) immediately after the first \(\bigl[{2\atop0}\bigr]\) in the last section of PF′, then cycle back the last section to be the first in the domino sequence.

 (3)Say \(\mathit{PF}'\in\mathcal{PF}_{J1}([p_{2},\ldots, p_{k},q])\) for a q⊨p _{1}−1>0

Let \(\operatorname{last}(\mathit{PF}')\) be the domino sequence obtained by removing the first k−1 sections from the domino sequence of PF′.

Modify \(\operatorname{last}(\mathit{PF}')\) by inserting a \(\bigl[{1\atop0}\bigr] \) immediately after its first \(\bigl[{2\atop0}\bigr]\) .

For a≥1 replace, in \(\operatorname{last}(\mathit{PF}')\) , each pair \(\bigl[{1\atop a1}\bigr] \bigl[{2\atop a}\bigr]\) by the pair \(\bigl[{2\atop a}\bigr] \bigl[{1\atop a}\bigr]\) .
(note that for this to put a big car on top of a big car we must have a \(\bigl[{2\atop a1}\bigr]\) preceding the \(\bigl[{1\atop a1}\bigr]\), but that \(\bigl[{2\atop a1}\bigr]\) will also be replaced either by this step or by the next steps)

For a≥1 replace each \(\bigl[{2\atop a}\bigr]\) preceded by a \(\bigl[{1\atop a}\bigr]\) in \(\operatorname{last}(\mathit{PF}')\) by \(\bigl[{2\atop a+1}\bigr]\)

Replace each \(\bigl [{2\atop0}\bigr]\) , except the first by a \(\bigl[{2\atop1}\bigr]\)
(note if a replaced \(\bigl[{2\atop0}\bigr]\) is preceded by a \(\bigl[{2\atop0}\bigr]\) then that \(\bigl[{2\atop0}\bigr]\) itself will also be replaced by \(\bigl[{2\atop1}\bigr]\))

The modified \(\operatorname{last}(\mathit{PF}')\) followed by the first k−1 sections of PF′ gives then the domino sequence of our target PF.

Since Φ moves EAST, by one cell, p _{1}−1 big cars it causes a loss of area equal to p _{1}−1. Thus the definition in (4.16) combined by the bijectivity of Φ proves the recursion in (4.9).
It remains to show equality in the base cases which, in view of the definition in (4.16) should be characterized by the absence of small cars.
Theorem 4.2
Proof
This completes of proof of (4.17). This was the last fact we need to establish the equality in (1.17). □
Remark 4.3
As we already mentioned, our definition of ndinv creates another puzzle. Indeed, the classical dinv can be immediately computed from the geometry of the parking function or directly from (1.6) which expresses it explicitly in terms of the two line array representation. For this reason we made a particular effort to obtain a nonrecursive construction of ndinv and in the best scenario derive form it an explicit formula similar to (1.6). However, our efforts yielded only a partially nonrecursive construction. In our original plan of writing we decided to include this further result even though in the end it yields a more complex algorithm for computing ndinv than from the original recursion. This was in the hope that our final construction may be conducive to the discovery of an explicit formula. It develops that during the preparation of this manuscript a new and better reason emerged for the inclusion of our final construction. It turns out that Angela Hicks and Yeonkyung Kim have very recently succeeded in discovering the desired explicit formula by a careful analysis of the combinatorial identities we are about to present. The results of Hicks–Kim will appear in a separate publication [13].
 (1)
The recursive construction will now consist of as many steps as there are dominos in the domino sequence
 (2)At each step the first domino of the first section is removed
 (a)
when we remove an \(\bigl[{s\atop0}\bigr]\) , the section is cycled to the end after it is processed as before
 (b)
when we remove a \(\bigl[{b\atop0}\bigr]\) , it is because the section consisted of a single big car domino.
 (a)
 (3)
The removal of an \(\bigl[{s\atop0}\bigr]\) contributes to ndinv the number of \(\bigl[{b\atop0}\bigr]\) ’s to its right minus one.
There are a few observations to be made about the effect of the cycling process. To begin note that when the domino sequence consists of a single section, no visible cycling occurs. However, even in this case, for accounting purposes, it is convenient to consider all of its dominos to have been cycled. With this provision, each domino in the original domino sequence will have an associated cycling number c that counts the number of times it has been cycled before it is removed.
Based on these observations, a step by step study of our recursive construction of ndinv led us to the following somewhat less recursive algorithm. It consists of two stages. In the first stage, the domino sequence is doctored and wrapped around a circle to be used in the second stage. The second stage uses circular motion to mimic the cycling of sections that takes place in the recursive procedure. To facilitate the understanding of the resulting algorithm we will illustrate each stage by applying it to the parking function in (4.21). More precisely we work as follows:
 Move each \(\bigl[{s\atop a}\bigr]\) in the domino sequence a places to its left and increase the area number by 1 of each domino \(\bigl[{b\atop a}\bigr]\) that is being bypassed. For instance the domino sections in (4.23) become$$ \everymath{\displaystyle} \begin{array}{@{}l} \left[ \left[ \begin{array}{c} 5\\0 \end{array} \right], \left[ \begin{array}{c} 2\\1 \end{array} \right], \left[ \begin{array}{c} 10\\2 \end{array} \right], \left[ \begin{array}{c} 7\\2 \end{array} \right], \left[ \begin{array}{c} 12\\0 \end{array} \right] \right], \qquad\left[ \left[ \begin{array}{c} 4\\0 \end{array} \right], \left[ \begin{array}{c} 1\\1 \end{array} \right], \left[ \begin{array}{c} 9\\2 \end{array} \right], \left[ \begin{array}{c} 6\\2 \end{array} \right], \left[ \begin{array}{c} 11\\0 \end{array} \right] \right], \\[4mm] \left[ \left[ \begin{array}{c} 3\\0 \end{array} \right], \left[ \begin{array}{c} 8\\1 \end{array} \right], \left[ \begin{array}{c} 13\\0 \end{array} \right] \right]. \end{array} $$(4.24)

Next wrap the resulting sequence clockwise around a circle with positions marked by a “∘”

Set ndinv=0 and set the auxiliary parameter c to 1.

Mark the first domino by changing its “∘” to a “•”.

Cycling clockwise from the first domino to the bar find the first \(\bigl[{b\atop0}\bigr]\) , call it “endsec”.

Cycling clockwise from endsec to the bar add 1 to ndinv each time we meet a \(\bigl[{b\atop0}\bigr]\) .

cycling clockwise from the last endsec mark the first unmarked domino

if in so doing the bar is crossed add 1 to c.

for each encountered unmarked \(\bigl[{b\atop a}\bigr]\) add 1 to ndinv provided a<c if the bar is not crossed or a<c+1 after the bar is crossed
Remark 4.4
We will not include a proof of the validity of this second algorithm, since A.S. Hicks and Y. Kim, using their discoveries, are able to provide in [13] a much simpler and more revealing validity argument than we can offer with our present tools. Here it should be sufficient to acknowledge that the auxiliary domino sequence resulting from Phase I together with the c statistic constructed in Phase II have ultimately been put to such beautiful use in subsequent work.
References
 1.Bergeron, F., Garsia, A.M.: Science fiction and Macdonald’s polynomials. In: Algebraic Methods and qSpecial Functions, Montréal, QC, 1996. CRM Proc. Lecture Notes, vol. 22, pp. 1–52. Am. Math. Soc., Providence (1999) Google Scholar
 2.Bergeron, F., Garsia, A.M., Haiman, M., Tesler, G.: Identities and positivity conjectures for some remarkable operators in the theory of symmetric functions. Methods Appl. Anal. 6, 363–420 (1999) MathSciNetzbMATHGoogle Scholar
 3.Garsia, A.M., Haglund, J.: A proof of the q,tCatalan positivity conjecture. Discrete Math. 256, 677–717 (2002) MathSciNetzbMATHCrossRefGoogle Scholar
 4.Garsia, A., Haiman, M.: Some natural bigraded modules and the q,tKostka coefficients. Electron. J. Comb. 3, Res. Paper 24 (1996) MathSciNetGoogle Scholar
 5.Garsia, A.M., Haiman, M.: A remarkable q,tCatalan sequence and qLagrange inversion. J. Algebr. Comb. 5(3), 191–244 (1996) MathSciNetzbMATHCrossRefGoogle Scholar
 6.Garsia, A., Haiman, M., Tesler, G.: Explicit plethystic formulas for the Macdonald q,tKostka coefficients. Séminaire Lotharingien de Combinatoire B42m (1999), 45 pp. Google Scholar
 7.Garsia, A.M., Xin, G., Zabrocki, M.: Hall–Littlewood operators in the theory of parking functions and diagonal harmonics. Int. Math. Res. Not. V. 11 (2011) Google Scholar
 8.Haglund, J.: A proof of the q,tSchröder conjecture. Int. Math. Res. Not. 11, 525–560 (2004) MathSciNetCrossRefGoogle Scholar
 9.Haglund, J.: The q,tCatalan Numbers and the Space of Diagonal Harmonics. AMS University Lecture Series, vol. 41 (2008), 167 pp. Google Scholar
 10.Haglund, J., Haiman, M., Loehr, N., Remmel, J.B., Ulyanov, A.: A combinatorial formula for the character of the diagonal coinvariants. Duke Math. J. 126, 195–232 (2005) MathSciNetzbMATHCrossRefGoogle Scholar
 11.Haglund, J., Morse, J., Zabrocki, M.: A compositional shuffle conjecture specifying touch points of the Dyck path. Can. J. Math. 64(4), 822–844 (2012). doi: 10.4153/CJM20110784 MathSciNetzbMATHCrossRefGoogle Scholar
 12.Haiman, M.: Hilbert schemes, polygraphs, and the Macdonald positivity conjecture. J. Am. Math. Soc. 14, 941–1006 (2001) MathSciNetzbMATHCrossRefGoogle Scholar
 13.Hicks, A.S., Kim, Y.: An explicit formula for the new “dinv” statistic for compositional “2shuffle” parking functions, to appear Google Scholar
 14.Macdonald, I.G.: Symmetric Functions and Hall Polynomials, 2nd edn. Oxford Mathematical Monographs. The Clarendon Press/Oxford University Press, New York (1995) zbMATHGoogle Scholar
 15.Zabrocki, M.: UCSD Advancement to Candidacy Lecture Notes. http://www.math.ucsd.edu/~garsia/somepapers/