Polynomial approach to explicit formulae for generalized binomial coefficients
 665 Downloads
Abstract
We extend the polynomial approach to the hook length formula proposed in Károlyi et al. Adv Math 277:252–282 (2015) to several other problems of the same type, including the number of paths formula in the Young graph of strict partitions.
Keywords
Combinatorial Nullstellensatz Polynomial identities Multidimensional interpolation Generalized binomial coefficients Graded graphMathematics Subject Classification
05E05 05E40 20C30The multivariate polynomial interpolation, or, in other words, the explicit form of Alon’s Combinatorial Nullstellensatz [9, 13], recently proved to be powerful in proving polynomial identities: in [9] it was used for the direct short proof of Dyson’s conjecture, later generalized in [10] for the equally short proof of the qversion of Dyson’s conjecture, then after additional combinatorial work it allowed to prove identities of Morris, Aomoto, Forrester (the last was open), their common generalizations, both in classical and qversions.
Here we outline how this method works in classical theory of symmetric functions, “selfproving” polynomial identities corresponding to the counting paths in the Young graph and “strict Young graph”, or, in other words, counting dimensions of linear and projective representations of symmetric groups.
The interpolationbased approach to symmetric functions has been earlier developed by Olshanski, Borodin, Okounkov, Vershik, Regev [5, 14, 15, 16] and others. It may sound speculatively, but according to author’s opinion, the novelty of my/author’s approach consists in considering arbitrary polynomials (rather antisymmetric, than symmetric) as functions of usual points and applying general facts about general (not symmetric) polynomials instead of considering “symmetric functions as functions of partitions” (formulation from [5]).
We start with a general framework of paths in graded graphs.
1 Graded graphs
Let G be a \(\mathbb {Z}\)graded countable directed graph with the vertex set Open image in new window , directed edges \((u,v)\in E(G)\) joining vertices of the consecutive levels \(u\in V_i\) and \(v\in V_{i+1}\) for some integer i. In what follows both indegrees and outdegrees of all vertices are bounded. This implies that the number \(P_{G}(v,u)\) of directed paths from v to u is finite for any two fixed vertices v, u. This number is known as the generalized binomial coefficient (usual binomial coefficient appears for the Pascal triangle).
Fix a positive integer k. Our examples are induced subgraphs of the lattice \(\mathbb {Z}^k\). Vertices are graded by the sum of coordinates, edges correspond to increasing of any coordinate by 1. That is, indegree of any vertex equals its outdegree and equals k. It is convenient to identify the vertex set V with the monomials Open image in new window , \((c_1,\dots ,c_k)\in \mathbb {Z}^k\). Then any edge corresponds to multiplying of monomial by some variable \(x_i\). Finite linear combinations of elements of V (with, say, rational coefficients, it is not essential hereafter) form a ring Open image in new window of Laurent polynomials in variables \(x_1,\dots ,x_k\). Not necessary finite linear combinations form a Open image in new window module which we denote by \(\Phi \). For monomial \(v\in V\) and \(\varphi \in \Phi \) we denote by Open image in new window the coefficient of v in the series \(\varphi \).
Define the minimum of two monomials \(u=\prod x_i^{a_i}\) and \(v=\prod x_i^{b_i}\) as Open image in new window . If Open image in new window we say that the monomial v majorates u.

Multidimensional Pascal graph \(\mathcal {P}_k\). This is a subgraph of \(\mathbb {Z}^k\) formed by the vectors with integral nonnegative coordinates (or, in the monomial language, by monomials having all variables in nonnegative power). For \(k=2\) this graph is isomorphic to the Pascal triangle.

Restricted Young graph \(\mathcal {Y}_k\). This is a subgraph of the Pascal graph formed by vectors \((c_1,\dots ,c_k)\) with strictly increasing nonnegative coordinates, \(0\leqslant c_1<c_2<\cdots <c_k\). The vertices of \(\mathcal {Y}_k\) may be identified with the Young diagrams having at most k rows (to any vertex \((c_1,\dots ,c_k)\) of \(\mathcal {Y}_k\) we assign a Young diagram with rows (to any vertex \((c_1,\dots ,c_k)\) of \(\mathcal {Y}_k\) we assign a Young diagram with lengths of rows \(c_1\leqslant c_21\leqslant \cdots \leqslant c_k(k1)\)). Edges correspond to addition of boxes. The usual Young graph has all Young diagrams as vertices, but when we count number of paths between two diagrams we may always restrict ourselves to \(\mathcal {Y}_k\) with large enough k.

Graph of strict partitions \(\mathcal {SY}_k\). This is a subgraph of the Pascal graph formed by vectors \((c_1,\dots ,c_k)\) with nonstrictly increasing nonnegative coordinates \(0\leqslant c_1\leqslant c_2\leqslant \cdots \leqslant c_k\) satisfying the following condition: if \(c_i=c_{j}\) for \(i\ne j\), then \(c_i=0\). To any vertex \((c_1,\dots ,c_k)\) we may assign a Young diagram with lengths of rows \(c_1,\dots ,c_k\). So, this diagram has at most k (nonempty) rows and they have distinct lengths.
Theorem 1.1
 (i)
 (ii)
if \(v'\in V(G)\) and \(\deg v=\deg v'\), then Open image in new window ;
 (iii)
if \(w\notin V(G)\), but \(x_iw\in V(G)\) for some \(x_i\), then Open image in new window .
Proof
Remark 1.2
In other words, both parts of (1) are fundamental solutions of the Laplace equation on the part of our graph starting from the level of v.
Theorem 1.1 leads to a natural question: for which labeled graphs (G, u) there exists a function \(\varphi \) such that conditions (i)–(iii) are satisfied? We do not know a full answer. The following statement is at least general enough to cover all examples of this paper.
Theorem 1.3
 (i)
(minimumclosed set) if \(u,w\in V\), then also Open image in new window .
 (ii)
(coordinate convexity) if \(u,w\in V\), \(w=ux_i^m\) for some index i and positive integer m, then also \(ux_i^s\in V\) for all \(0\leqslant s\leqslant m\).
Proof
Call a monomial \(u\in \mathbb {Z}^k\) special if either \(u\in V\), \(\deg u=\deg v\) (in particular, v itself is special), or \(\deg u\geqslant \deg v\), \(u\notin V\), \(x_iu\in V\) for some variable \(x_i\). It suffices to prove that there exists a function \(\varphi \) homogeneous of degree \(\deg v\) with any prescribed values of Open image in new window for all special u. This is a linear system on coefficients of \(\varphi \).
By replacing V with \(v^{1}V\) we may suppose that \(v=1\). For any special monomial u define its son S(u) as follows: if \(\deg u=0\), then \(S(u)=0\), if \(u\notin V\), \(x_iu\in V\), then \(S(u)=x_i^{\deg u} u\) (if different indexes i satisfy \(x_iu\in V\), choose any).
If \(u\ne w\) are two special monomials and w majorates S(u), then \(\deg w>\deg u\). Indeed, assume that on the contrary \(\deg w\leqslant \deg u\). First, if \(\deg u=0\), then \(\deg w=0\) and \(w=S(u)=u\), a contradiction. Thus \(d=\deg u>0\), we may suppose that \(x_1u\in V\), \(S(u)=x_1^{d} u\). Denote Open image in new window , \(a_i\geqslant 0\), \(d\geqslant a_1+\cdots +a_k\), and either \(w\in V\) or \(x_iw\in V\) for some i. Since \(w\ne u\), we have \(d>a_1\). Denote Open image in new window if \(x_iw\in V\), and Open image in new window if \(w\in V\). Then \(u_0=x_1^{a_1d+\varepsilon }u\), where Open image in new window , and \(u_0\in V\) since V is a minimumclosed set. The coordinate convexity implies that if both \(u_0,x_1u\) belong to V, so does u. A contradiction.
Now we start to solve our linear system for coefficients of \(\varphi \). For any special monomial u we have a linear relation on the monomials of degree 0 majorated by u. Between them there is a monomial S(u), and it does not appear in relations corresponding to special monomials \(w\ne u\) with \(\deg w\leqslant \deg u\). It allows to fix coefficients of \(\varphi \) in the appropriate order (by increasing the degree of u) and fulfil all our relations. \(\square \)
2 Observation on polynomials
Recall the Combinatorial Nullstellensatz of Alon [1].
Theorem 2.1
For verifying polynomial identities which allow to calculate coefficients we use the following observation in the spirit of the Combinatorial Nullstellensatz.
Let K be a field and Open image in new window , \(i=1,\dots ,k\), be its subsets of size \(A_i=n+1\).
Observation 2.2
Proof
The number of points in \(\Delta \) equals exactly the dimension of the space of polynomials with degree at most n in k variables. Hence it suffices to check either existence or uniqueness of the polynomial with degree at most n and with prescribed values on \(\Delta \). Both tasks are easy as we may see:
Uniqueness. It suffices to prove that the polynomial \(f(x_1,\dots ,x_k)\) with degree at most n, which vanishes on \(\Delta \), identically equals 0. Assume the contrary. Let Open image in new window be a highest degree term in f. The set \(\Delta \) contains the product Open image in new window , where Open image in new window , \(B_i=t_i+1\). By the Combinatorial Nullstellensatz, f cannot vanish on \(\prod B_i\). A contradiction. \(\square \)
We use notations \({x}^{\underline{n}}=x(x1)\cdots (xn+1)\) and \(\left( {\begin{array}{c}x\\ n\end{array}}\right) ={x}^{\underline{n}}/n!\).
Remark 2.3
Some other notations, including Open image in new window are used for falling factorials. I/author prefer the Capelli–Toscano notation, popularized by Knuth, see his arguments in [12]. The author finds it quite intuitive, particularly in the context of this paper.
The following particular case of interpolation on \(\Delta _k^n\) appears to be useful.
Lemma 2.4
Proof
It suffices to check the equality for values on \(\Delta _k^n\). Both parts vanish on \(\Delta _k^{n1}\). If \((x_1,\dots ,x_k)\) is a point on Open image in new window , i.e. \(\sum x_i=n\), then all summands on the right vanish except (possibly) the summand with \(c_i=x_i\) for all i, and its value just equals \(f(c_1,\dots ,c_k)=f(x_1,\dots ,x_k)\), i.e. the value of LHS at the same point, as desired. \(\square \)
We start our series of applications of Observation 2.2 along with the multinomial version of the Chu–Vandermonde identity.
2.1 Example: Chu–Vandermonde identity and multinomial coefficient
3 Hook length formula
We start with a polynomial identity which is in a sense similar to the Chu–Vandermonde identity (3).
Theorem 3.1
Proof
By Lemma 2.4, it suffices to check that LHS vanishes on \(\Delta _k^{n+k(k1)/21}\). Let \(x_i\) be nonnegative integers and \(\sum x_i<n+k(k1)/2\). If \(\sum x_i<k(k1)/2\) then some factor \(x_ix_j\) vanishes, otherwise \(y=\sum x_ik(k1)/2\) is nonnegative and \(y<n\), hence Open image in new window .\(\square \)
Corollary 3.2
Proof
Take \(\varphi =\prod _{i<j}(x_jx_i)\) and apply Theorem 1.1. Conditions (i) and (ii) follow from (5) with \(n=0\) (actually, this is just the Vandermonde determinant formula). For checking (iii), note that such \(w=\prod x_i^{n_i}\) satisfies either \(n_i<0\) for some i or \(n_i=n_j\) for some i, j. In both cases (5) yields that the corresponding coefficient vanishes. \(\square \)
Note that our proof of (6) does not use the Multinomial Theorem, but is proved in the same way and the proof is almost equally short. Identity (5) appears also in the important for the development of the polynomial method paper [2], where it is used for appropriate application of the Combinatorial Nullstellensatz, while we show how it may be proved by the (explicit version of) Combinatorial Nullstellensatz.
In the rest part of this section we explain the relation of (6) with hook lengths of the Young diagram, this is mostly for the sake of completeness.
Recall that the graph \(\mathcal {Y}_k\) may be viewed as the graph of Young diagrams having at most k rows. For a vertex \(v\in \mathcal {Y}_k\), Open image in new window , the corresponding diagram \(\lambda (v)\) has k (possibly empty) rows with lengths \(n_1\leqslant n_21\leqslant \cdots \leqslant n_k(k1)\). Edges of the graph \(\mathcal {Y}_k\) correspond to adding boxes, and paths correspond to the skew standard Young tableaux: for any path with, say m edges, put numbers \(1,2,\dots ,m\) in the corresponding adding boxes. In this language, expression (6) counts the number of standard Young tableaux of the shape \(\lambda (v)\). Assuming \(n_1>0\) (i.e. the number of rows equals k), we may interpret parameters \(n_1,\dots ,n_k\) as hook lengths of k boxes in the first column. Recall that (now specify that the largest column in the Young diagram is the leftmost and the largest row is the lowest) a hook of a box X in the Young diagram is a union of X; all boxes in the same column which are higher than X; all boxes in the same row which are on the right to X.
Claim
Proof
Assume that boxes a, b of the Young diagram lie in the same row and a, c in the same column. Let d be a box such that abdc is a rectangle. If d belongs to the diagram then \(h(a)<h(b)+h(c)\), otherwise \(h(a)>h(b)+h(c)\). Hence we always have the inequality \(h(a)\ne h(b)+h(c)\). Now \(n_1,\dots ,n_k\) are hook lengths of boxes in the first column. In the ith row there are Open image in new window boxes, and their hook lengths are distinct numbers from 1 to \(n_i\), with \(i1\) values excluded, and those excluded values are \(n_in_1, n_in_2, \dots , n_in_{i1}\), by the inequality. It remains to multiply over \(i=1,2,\dots ,k\). \(\square \)
The above claim allows to formulate Corollary 3.2 in the form of the hook length formula [6].
Theorem 3.3
(hook length formula) The number of the standard Young tableaux of a given shape \(\lambda \) with n boxes equals \(n!/\prod _{\square } h(\square )\), where the product is taken over all boxes of \(\lambda \).
4 Skew Young tableaux
Here we get generalizations of Theorem 3.1, corresponding to counting paths between two arbitrary vertices of \(\mathcal {Y}_k\) (i.e. the number of skew Young tableaux of a given shape).
Theorem 4.1
Proof
Due to Lemma 2.4, it suffices to check that LHS vanishes on \(\Delta _k^{n+m1}\). Fix a point \((x_1,\dots ,x_k)\in \Delta _k^{n1}\). Let \(y_1\leqslant \dots \leqslant y_k\) be the increasing permutation of \(x_i'\). If \(y_i<m_i\) for some i, then the matrix Open image in new window is singular as it has Open image in new window minor of zeros, and the matrix Open image in new window is therefore singular too. If \(y_i\geqslant m_i\) for all i, then denoting \(y=\sum x_im=\sum (y_im_i)\geqslant 0\) we have \(0\leqslant y<n\), hence Open image in new window . \(\square \)
Corollary 4.2
Proof
Take \(\varphi =a_{m_1,\dots ,m_k}(x_1,\dots ,x_k)\) and apply Theorem 1.1. Conditions (i) and (ii) follow from (7) with \(n=0\) (or from expanding the determinant). For checking (iii), note that such \(w=\prod x_i^{n_i}\) satisfies either \(n_i<0\) for some i or \(n_i=n_j\) for some i, j. In both cases (7) yields that the corresponding coefficient vanishes. \(\square \)
Remark 4.3
This formula for the number of paths between two arbitrary vertices of the Young graph appeared in [15, Theorem 8.1], see also [16]. Recently it appeared in the context of additive combinatorics in [3, Lemma 4], [4].
The value \(b_{m_1,\dots ,m_k}(n_1,\dots ,n_k)\) has a combinatorial interpretation following from the Lindström–Gessel–Viennot Lemma: up to a multiple \(\prod m_i!\) it is a number of semistandard Young tableaux of a given shape and content. See details in [7].
A.M. Vershik pointed out that similar results are known for the graph of strict diagrams. It also may be included in our framework.
5 Strict diagrams
For counting the number of paths in the graph \(\mathcal {SY}_k\) we need series which are not polynomials.
Let \(x_1,\dots ,x_k\) be variables (as before). Consider the set \(\mathcal {M}\) of rational functions in those variables with denominator Open image in new window . Expand such functions in Laurent series in \(x_1,x_2/x_1,x_3/x_2,\dots ,x_k/x_{k1}\) (i.e., Open image in new window for \(i<j\)). Define the value of such function at a point \((c_1,\dots ,c_k)\) with nonnegative coordinates as follows: if coordinates are positive, just substitute them in the function, if some coordinates vanish, replace them by positive numbers Open image in new window (in such order), and let t tend to Open image in new window . What we actually need is that for \(i<j\) the value Open image in new window with \(x_i=x_j=0\) equals 0. Each function \(f\in \mathcal {M}\) may be expanded as \(f=P[f]+Q[f]\), where P[f] is a polynomial in \(x_1,\dots ,x_k\), and in Q[f] each term \(\prod x_i^{c_i}\) contains at least one variable \(x_i\) in a negative power \(c_i<0\). We say that P[f] is the polynomial component of f and Q[f] is the antipolynomial component of the function \(f(x_1,\dots ,x_k)\).
Lemma 5.1
 (i)
its antipolynomial component \(Q[f_n]\) vanishes on the standard simplex \(\Delta _k^n\),
 (ii)
if \(c_1,\dots ,c_k\) are integers such that \(c_j<0\), \(c_i\geqslant 0\) for \(i=j+1,\dots ,k\), then \(\bigl [\prod x_i^{c_i}\bigr ]f_n=0\).
Proof
For proving (ii), note that we should have \(x_j=\xi _s\) for some s, but if \(\xi _s\) is taken in negative power, then \(\zeta _s=x_{b_s}\) must be taken in positive power and \(b_s>j\). \(\square \)
Modulo this lemma everything is less or more the same as in previous sections.
Theorem 5.2
Proof
Both parts are polynomials of degree at most n, thus it suffices to check that their values at each point \((c_1,\dots ,c_k)\in \Delta _k^n\) are equal. The polynomial component of \(f_n(x_1,\dots ,x_k)\) takes the same values on \(\Delta _k^n\) as the function \(f_n\). Thus, by Lemma 2.4, it suffices to check that \(f_n\) vanishes on \(\Delta _k^{n1}\). But already the factor Open image in new window vanishes on \(\Delta _k^{n1}\). \(\square \)
Corollary 5.3
Corollary 5.4
Proof
Apply Theorem 1.1 to the function Open image in new window . Then the result directly follows from Corollary 5.3, so it suffices to check all conditions of Theorem 1.1. Condition (i) follows from Corollary 5.3 for \(n=0\) (or just from common sense). In condition (ii) there is nothing to check since 1 is the unique vertex of \(\mathcal {SY}_k\) with degree 0. Let us check condition (iii). Assume that \(w=\prod x_i^{c_i}\) is such that \(x_iw\) is a vertex of \(\mathcal {SY}_k\) but w is not. There are two cases: either all coordinates of w are nonnegative and \(c_j=c_l>0\) for some j, l, or Open image in new window , \(c_j= 0\) for all \(j\geqslant i\). In the first case apply Corollary 5.3, in the second case apply statement (ii) of Lemma 5.1. \(\square \)
6 Skew strict Young tableaux
Here we give an identity corresponding to the formula [8, 17] for the number of paths between any two vertices of the graph \(\mathcal {SY}_k\), or, in other words, for the number of strict skew Young tableaux of a given shape.
Let \(v=\prod _{i=1}^k x_i^{m_i}\), \(m_1>m_2>\cdots >m_{\ell }>m_{\ell +1}=0=\cdots =m_k\), be a vertex of \(\mathcal {SY}_k\), \(u=\prod x_i^{n_i}\) be another vertex of \(\mathcal {SY}_k\) and \(n_i\geqslant m_i\) for all i (thus there exists some path from u to v). Denote \(m=\sum m_i\), \(n=\sum n_i\).
Theorem 6.1
 (i)
The antipolynomial component Open image in new window vanishes on the simplex \(\Delta _k^n\).
 (ii)The polynomial component of g has the expansion
 (iii)The number of paths from \(v=\prod x_i^{m_i}\) to \(u=\prod x_i^{n_i}\) equals$$\begin{aligned} \frac{(nm)!}{\prod n_i!} \cdot \varphi _v(n_1,\dots ,n_k). \end{aligned}$$
Proof
Note that as in Lemma 5.1 we may also conclude that if \(c_1,\dots ,c_k\) are integers such that \(c_j<0\), \(c_i\geqslant 0\) for \(i=j+1,\dots ,k\), then Open image in new window .
(ii) The values of Open image in new window on \(\Delta _k^n\) are the same as values of g. Any summand of the above expansion for g vanishes on \(\Delta _k^{n1}\). Thus it suffices to use Lemma 2.4.
(iii) Apply Theorem 1.1 to the function \(\varphi _v\) (or its leasing part, it is a matter of taste). It suffices to check all conditions of Theorem 1.1. Conditions (i) and (ii) follow from the above expansion of g (with \(n=0\)). There are exactly Open image in new window permutations with \(y_i=x_i\), \(i=1,\dots ,\ell \), for each of them we get coefficient 1 in the monomial v and coefficient 0 in other monomials of degree m. For other permutations we do not get nonzero coefficients in monomials which are vertices of \(\mathcal {SY}_k\). Let us check condition (iii). Assume that \(w=\prod x_i^{c_i}\) is such that \(x_iw\) is a vertex of \(\mathcal {SY}_k\) but w is not. There are two cases: either all coordinates of w are nonnegative and \(c_j=c_l>0\) for some j, l, or \(c_i=1\), \(c_j= 0\) for all \(j\geqslant i\). In the first case apply identity (8), in the second case apply the above remark after the proof of part (i) of the theorem. \(\square \)
7 Concluding remarks
 (a)
\(\varphi =\prod x_i^{m_i}\), \(v=\prod x_i^{m_i}\), G is the multidimensional Pascal graph \({\mathcal P}_k\).
 (b)
Open image in new window , \(m_1<m_2<\dots <m_k\), \(v=\prod x_i^{m_i}\), G is the Young graph \({\mathcal Y}_k\) formed by strictly increasing sequences.
 (c)
Open image in new window , v is the origin, \(G={\mathcal SY}_k\) is the graph of strict Young diagrams. When v is not the origin, the corresponding function \(\varphi _v\) should be multiplied by the Okounkov polynomial, as described in Sect. 6.
Another question which has to be answered is how identities with falling factorials are related with asymptotics of dimensions.
Notes
Acknowledgments
This work originated from collaboration with Gyula Károlyi, Zoltán Nagy and Vladislav Volkov. The idea to study the graph of strict diagrams in the same spirit belongs to Anatoly Vershik. To all of them the author is really grateful. He is also grateful to the referees for useful suggestions and especially for pointing out several important papers on the subject.
References
 1.Alon, N.: Combinatorial Nullstellensatz. Combin. Probab. Comput. 8(1–2), 7–29 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
 2.Alon, N., Nathanson, M.B., Ruzsa, I.: The polynomial method and restricted sums of congruence classes. J. Number Theory 56(2), 404–417 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
 3.Balandraud, É.: An addition theorem and maximal zerosum free sets in \(\mathbb{Z}/p\mathbb{Z}\). Israel J. Math. 188, 405–429 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
 4.Balandraud, É.: Erratum to: “An addition theorem and maximal zerosum free sets in \({\mathbb{Z}}/p{\mathbb{Z}}\)”. Israel J. Math. 192(2), 1009–1010 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
 5.Borodin, A., Olshanski, G.: Harmonic functions on multiplicative graphs and interpolation polynomials. Electron. J. Combin. 7, R28 (2000)MathSciNetzbMATHGoogle Scholar
 6.Frame, J.S., Robinson, G. de B., Thrall, R.M.: The hook graphs ofthe symmetric groups. Canad. J. Math. 6, 316–324 (1954)Google Scholar
 7.Fulmek, M.: Viewing determinants as nonintersecting lattice paths yields classical determinantal identities bijectively. Electron. J. Combin. 19(3), P21 (2012)MathSciNetzbMATHGoogle Scholar
 8.Ivanov, V.N.: Dimensions of skewshifted young diagrams and projective characters of the infinite symmetric group. J. Math. Sci. (N. Y.) 96(5), 3517–3530 (1999)CrossRefGoogle Scholar
 9.Karasev, R.N., Petrov, F.V.: Partitions of nonzero elements of a finite field into pairs. Israel J. Math. 192(1), 143–156 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
 10.Károlyi, G., Nagy, Z.L.: A simple proof of the Zeilberger–Bressoud \(q\)Dyson theorem. Proc. Amer. Math. Soc. 142(9), 3007–3011 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
 11.Károlyi, G., Nagy, Z.L., Petrov, F.V., Volkov, V.: A new approach to constant term identities and Selbergtype integrals. Adv. Math. 277, 252–282 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
 12.Knuth, D.E.: Two notes on notation (1992). arxiv:math/9205211
 13.Lasoń, M.: A generalization of combinatorial Nullstellensatz. Electron. J. Combin. 17(1), N32 (2010)MathSciNetzbMATHGoogle Scholar
 14.Okounkov, A.: On Newton interpolation of symmetric functions: a characterization of interpolation Macdonald polynomials. Adv. in Appl. Math. 20(4), 395–428 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
 15.Okounkov, A., Olshanski, G.: Shifted Schur functions. St. Petersburg Math. J. 9(2), 73–146 (1997)MathSciNetzbMATHGoogle Scholar
 16.Olshanski, G., Regev, A., Vershik, A.: Frobenius–Schur Functions. In: Joseph, A., Melnikov, A., Rentschler, R. (eds.) Studies in Memory of Issai Schur. Progress in Mathematics, vol. 210, pp. 251–299. Birkhäuser, Boston (2003)Google Scholar
 17.Thrall, R.M.: A combinatorial problem. Michigan Math. J. 1(1), 81–88 (1952)MathSciNetCrossRefzbMATHGoogle Scholar