The symmetric functions catalog

#\(q\)-analogs

We use the \(q\)-integer, \(q\)-factorial, and \(q\)-binomial coefficient: \[[n]_q \coloneqq \frac{1-q^n}{1-q}, \quad [n]_q! \coloneqq [n]_q [n-1]_q \dotsm [2]_q [1]_q, \quad \qbinom{n}{k}_q \coloneqq \frac{[n]_q!}{[k]_q! [n-k]_q!}.\] Furthermore, for a composition \(\alpha \vDash n,\) the \(q\)-multinomial coefficient is \[\qbinom{n}{\alpha}_q \coloneqq \frac{[n]_q!}{[\alpha_1]_q! [\alpha_2]_q! \dotsm [\alpha_\ell]_q!}\]

We have \[\sum_{\pi \in \symS_n} q^{\maj(\pi)} = \sum_{\pi \in \symS_n} q^{\inv(\pi)} = [n]_q!.\] These identities are true for charge and cocharge as well. Combinatorial statistics on permutations with this distribution are called Mahonian.

Note that \[\qbinom{n}{k}_q = \sum_{w \in BW(n,k)} q^{\maj(w)} = \sum_{w \in BW(n,k)} q^{\inv(w)}\] where \(BW(n,k)\) is the set of binary words of length \(n\) with exactly \(k\) ones. This generalizes as follows. Let \(W(n,\alpha)\) denote the set of words of length \(n,\) with weight \(\alpha.\) Then we have the following identity for the \(q\)-multinomial coefficients: \[\qbinom{n}{\alpha}_q = \sum_{w \in W(n,\alpha)} q^{\maj(w)} = \sum_{w \in W(n,\alpha)} q^{\inv(w)}.\] For proofs, see the lecture notes by D. Foata and G.-N. Han [FH11].

#\(q\)-derangements and shuffles

Let \(D_n \subseteq \symS_n\) be the set of derangements, that is, permutations with no fixed points. The \(q\)-derangement number is \[d_n(q) \coloneqq \sum_{\sigma \in D_n} q^{\maj(\sigma)}.\] I. M. Gessel and C. Reutenauer derived the following formula from their quasisymmetric generating function for permutations with fixed cycle structure and descent set [GR93]. A combinatorial proof was later given by M. L. Wachs [Wac89]. A recent direct proof using decorated permutations is due to K. Q. Ji [Ji26].

The key refinement in Wachs’s proof is to count permutations by their derangement part. If \(\pi \in \symS_n,\) then \(\operatorname{dp}(\pi)\) is the reduction of the subword of non-fixed points of \(\pi.\) For example, \[\operatorname{dp}(1\,5\,3\,7\,6\,2\,9\,8\,4)=3\,5\,4\,1\,6\,2.\]

Summing over all derangements \(\sigma \in D_k\) gives \[[n]_q! = \sum_{k=0}^n \qbinom{n}{k}_q d_k(q),\] and the formula for \(d_n(q)\) follows by \(q\)-binomial inversion.

The shuffle identity used in this proof is due to A. M. Garsia and I. M. Gessel [Thm. 3.1, GG79]. Let \(\alpha\) and \(\beta\) be two words with disjoint letter sets. A shuffle of \(\alpha\) and \(\beta\) is an interleaving that preserves the internal order of the letters in each word. Write \(\operatorname{Sh}(\alpha,\beta)\) for the set of such shuffles.

We define the Pochhammer symbol as \((a)_n \coloneqq a(a+1)\dotsm (a+n-1)\) and the \(q\)-Pochhammer symbol as \[(a;q)_n \coloneqq \prod_{j=0}^{n-1}(1-a q^j).\]

#\(q\)-Lucas theorem

The \(q\)-Lucas theorem first appeared in [Eq. (1.2.4), Oli65] and later in [Prop. 2.2, De82]. It was perhaps known by Gauß already. A combinatorial proof was first given by V. Strehl [Str81].

Let \(n=n_1 d + n_0\) and \(k=k_1 d + k_0,\) where \(0\leq n_0, k_0 \lt{} d.\) We have \[\qbinom{n}{k}_q \equiv \binom{n_1}{k_1} \qbinom{n_0}{k_0}_q \mod \Phi_d\] where \(\Phi_d\) is the \(d^\thsup\) cyclotomic polynomial. In particular \[\qbinom{n}{k}_q = \binom{n_1}{k_1} \qbinom{n_0}{k_0}_q \text{ whenever } q=e^{2\pi i \frac{c}{d}}\] and \(\gcd(c,d)=1.\) A nice proof using cyclic sieving is given in [Sag92].

S. C. Billey and J. P. Swanson survey cyclotomic generating functions, whose roots are roots of unity or zero [BS23]. They discuss many examples from combinatorial representation theory, conditions for such polynomials to exist, and asymptotic normality results for their coefficients.

#Various \(q\)-identities

Cauchy’s \(q\)-binomial theorem states that \[\prod_{j=1}^n (1+yq^j) = \sum_{j=0}^n y^j q^{j(j+1)/2} \qbinom{n}{j}_q\]

The following is a very useful identity.

#Catalan numbers and Fuß–Catalan numbers

The MacMahon \(q\)-analog of the Catalan numbers is the following: \[\begin{aligned} \catalan(n;q) &= \frac{1}{[n+1]_q}\qbinom{2n}{n}_q \\ &= \frac{[2n]_q!}{[n+1]_q![n]_q! } \\ &= q^{-n} \qbinom{2n}{n}_q - \qbinom{2n}{n+1}_q \\ &= \qbinom{2n}{n}_q - q \qbinom{2n}{n-1}_q \end{aligned}\] Note that \[\catalan(n;q) = \sum_{w \in DP(n)} q^{\maj(w)}\] where \(DP(n)\) is the set of binary words of length \(2n\) with \(n\) ones such that the number of ones in any initial segment never exceeds the number of zeros in the same segment.

The \(q\)-analog of Fuß–Catalan numbers can be expressed as \[\begin{aligned} \catalan(n,m;q) &\coloneqq \frac{1}{[n]_q} \qbinom{(m+1)n}{n-1}_q \\ &= \frac{1}{[(m+1)n+1]_q}\qbinom{(m+1)n+1}{n}_q \\ &= \qbinom{(m+1)n}{n}_q - q \frac{[mn]_q}{[n]_q} \qbinom{(m+1)n}{n-1}_q. \end{aligned}\]

The rational Catalan numbers \(\catalan(a/b;q)\) are defined as \[\catalan(a/b;q)= \frac{1}{[a+b]_q} \qbinom{a+b}{a}_q.\] This is a \(q\)-analog of the number of paths from \((0,0)\) to \((b,a)\) staying weakly above the line \(y=\frac{a}{b}x.\) These objects admit cyclic sieving phenomena.

#q-Narayana numbers

The \(q\)-Narayana numbers are defined as

\[\begin{aligned} N(n,k;q) &\coloneqq \frac{q^{k(k-1)}}{[n]_q} \qbinom{n}{k}_q \qbinom{n}{k-1}_q \\ &= \frac{1}{[k]_q} \qbinom{n-1}{k-1}_q \qbinom{n}{k-1}_q \\ &= q^{k(k - 1)- n} \left(\qbinom{n - 1}{k - 1}_q \qbinom{n + 1}{k}_q - \qbinom{n}{k - 1}_q\qbinom{n}{k}_q \right). \end{aligned}\] Note that \(\catalan(n;q) = \sum_{k=1}^n N(n,k;q)\) and that \[\sum_{w \in DP(n)} q^{\maj(w)} t^{\text{peaks}(w)} = \sum_{k=1}^n t^k N(n,k;q).\] where the sum is over all Dyck paths.

In [Thm. 6, Br04], it is proved that \[N(n,k+1;q) = \schurS_{2^k}(q,q^2,\dotsc,q^{n-1}).\]

#\(q\)-identities and symmetric functions

#\(q\)-hook formula and Schur polynomials

We have the following symmetry under conjugation: \[f^{\lambda}(q) = q^{\binom{n}{2}} f^{\lambda'}(1/q).\]

For a short proof, see [BD09].

By using RSK, one can show that \[\sum_{\sigma \in \symS_n} t^{\maj(\sigma)} q^{\maj(\sigma^{-1})} = \sum_{\lambda \vdash n} \sum_{P,Q \in \SYT(\lambda)} t^{\maj(P)} q^{\maj(Q)}.\]

The Cauchy identity then gives that \[\sum_{n \geq 0} \frac{z^n}{(q)_n (t)_n} \sum_{\pi \in \symS_n} t^{\maj(\pi)} q^{\maj(\pi^{-1})} = \prod_{i,j \geq 0} \frac{1}{1-z q^i t^j}.\] Here, \((q)_n = (1-q)(1-q^2)\dotsb (1-q^n).\) This identity appears in [Eq. (7.117), Sta01]. Note that Stanley uses a different definition for \([n]_q!.\)

#Distribution with fixed number of descents

In [Kei18], some results regarding two-row Young tableaux refined by descents are given. Let \[f^{\lambda/\mu}_d(q) \coloneqq \sum_{\substack{ T \in \SYT(\lambda/\mu) \\ |\DES(T)|=d} } q^{\maj(T)}.\]

The same paper shows that for \(\lambda \vdash n\) and \(m \geq \lambda_1,\) \[f^{(m,\lambda)}_{n}(q) = q^{\binom{n+1}{2}}\schurS_{\lambda}(1,q,q^2,\dotsc,q^{m-1}).\]

In [Che20], the author studies the major-index distribution of so-called increasing tableaux, introduced by O. Pechenik [Pec14]. In particular, X. Chen provides nice \(q\)-binomial formulas in the case of two-row skew shapes.

Using properties of the Robinson–Schensted–Knuth correspondence or jeu de taquin, one can show that for any \(D\subseteq [n-1],\) \[\sum_{\substack{T \in \SYT(\lambda/\mu) \\ \DES(T) = D}} q^{\maj(T)} = \sum_{\nu \vdash n} \sum_{\substack{T \in \SYT(\nu)\\ \DES(T) = D}} c^{\lambda}_{\mu\nu} q^{\maj(T)}.\] and \[\sum_{\substack{T \in \SYT(\lambda/\mu) \\ \DES(T) = D}} q^{\cocharge(T)} = \sum_{\nu \vdash n} \sum_{\substack{T \in \SYT(\nu)\\ \DES(T) = D}} c^{\lambda}_{\mu\nu} q^{\cocharge(T)}.\]

See also S. Pfannerer’s descent refinement of the Murnaghan–Nakayama rule [Pfa21] for evaluations of similar polynomials at roots of unity.

R. M. Adin and Y. Roichman introduce a cyclic descent framework for involutions interpreted as matchings [AR23]. They construct explicit cyclic descent extensions for involutions, standard Young tableaux, and Motzkin paths, and prove Schur positivity for the associated quasisymmetric functions.

#Evaluations at roots of unity

The following proposition generalizes the above identity. It is easy to prove using the identities for the principal specialization of Schur polynomials above.

The paper [DLT94] contains several evaluations of the transformed Hall–Littlewood polynomials at roots of unity. In particular, \(K_{\lambda/\nu,\mu^k}(\xi)\) is up to some sign equal to \(K^{(k)}_{\lambda/\nu,\mu},\) the number \(k\)-ribbon tableaux of shape \(\lambda/\nu\) and weight \(\mu.\)

See also [AK22, Alb23], where Lie group characters of types \(A,\) \(B,\) \(C,\) and \(D\) are evaluated at roots of unity.

See also [Kum22] and [Chap. I, §5, Ex. 24, Mac95]. A generalization of this identity for flagged Schur polynomials is given in [Kum23].

In [Alb23], one notes that the algebra homomorphism \(\phi_r\) that sends \(\completeH_j\) to \(\completeH_{j/r}\) if \(r \mid j,\) and \(0\) otherwise, corresponds to evaluating the symmetric function at \((x_i^{1/r}, \xi x_i^{1/r}, \xi^2 x_i^{1/r}, \dotsc, \xi^{r-1} x_i^{1/r})\) for each variable \(i.\) The above theorem by Lee–Oh can then be lifted to the symmetric functions level, see [Alb23].

#\(q\)-Pochhammer and basic hypergeometric series

When working with \(q\)-analogs, it is convenient to introduce some additional notation.

Recall that the \(q\)-Pochhammer symbol is \[(a;q)_n \coloneqq (1-a)(1-aq)\dotsm (1-aq^{n-1}), \qquad (a;q)_0 \coloneqq 1.\] This is also known as the \(q\)-shifted factorial. Using this notation, S. Formichella and A. Straub extend Gaussian binomial coefficients to negative integer arguments and prove corresponding negation, symmetry, and Lucas-type congruence identities [FS19]. From the definition, we have the relation \[(q^a;q)_{n+r} = (q^a;q)_{r} (q^{a+r};q)_n.\] By convention, \((a;q)_\infty \coloneqq \prod_{j \geq 0}(1-aq^j),\) so that \[(a;q)_n = \frac{(a;q)_\infty}{(aq^{n};q)_\infty}.\] This allows us to set \[(a;q)_{-n} \coloneqq \frac{1}{ (aq^{-n};q)_n } = \frac{ (-q/a)^n q^{\binom{n}{2}} }{ (q/a;q)_n }.\] One can then prove that \[(a q^{-n};q)_{n} = q^{-\binom{n}{2}} \left( -\frac{a}{q} \right)^n (q/a;q)_{n}.\]

We have the following identities: \[\begin{aligned} _q &= \frac{ (q;q)_m}{(1-q)(q;q)_{m-1}} = \frac{(q^{m};q)_\infty}{(1-q)(q^{m+1};q)_\infty} \\ [m]_q! &= \frac{ (q;q)_m}{(1-q)^m} \\ \qbinom{m}{r}_q &= \frac{(q^{m-r+1};q)_r}{(q;q)_r} = \frac{(q^{r+1};q)_\infty (q^{m-r+1};q)_\infty }{ (q^{m+1};q)_\infty (q;q)_\infty }. \end{aligned}\]

See the appendix of [GR04] for many more identities.

#\(q\)-binomial theorem

The \(q\)-binomial theorem can be reformulated using \(q\)-hypergeometric series. We have the identity \[\sum_{n=0}^\infty \frac{(a;q)_n}{(q;q)_n}z^n = \frac{(az;q)_\infty}{(z;q)_\infty}.\]

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