Product formulae for trace generating functions

for semistandard tableaux of unusual shapes^{†}

C. Krattenthaler

Fakultät für Mathematik der Universität Wien, Nordbergstraße 15, A-1090 Wien, Austria.

                 WWW: http://www.mat.univie.ac.at/~kratt

Abstract. Given a partition λ = (λ_{1}, …, λ_{r}), we show that the trace generating functions for semistandard tableaux of shapes λ/(0, 0, 1, 1, …) and λ/(0, 1, 2, 2, 2, …) are both given by closed product formulae.

An r-tuple λ = (λ_{1}, …, λ_{r}) with λ_{1} ≥ λ_{2} ≥ ⋯ ≥ λ_{r} ≥ 0 is called a partition. The components λ_{i} are called parts of the partition. Let λ be a partition and μ = (μ_{1}, μ_{2}, …, μ_{r}) an arbitrary r-tuple of non-negative integers with μ_{i} ≤ λ_{i} for i = 1, 2, …, r. A (semistandard) tableau τ of shape λ/μ is an array

\begin{matrix}τ_{1,μ_{1}+1} & τ_{1,μ_{1}+2} & . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . & τ_{1,λ_{1}} \\ τ_{2,μ_{2}+1} & τ_{2,μ_{2}+2} & . . . . . . . . & τ_{2λ_{2}} \\ ⋮ & . . . . . . . . . . . . & ... \\ τ_{r,μ_{r}+1} & … & τ_{rλ_{r}} \end{matrix}  (1)

of non-negative integers τ_{ij}, 1 ≤ i ≤ r, μ_{i} < j ≤ λ_{i}, such that the rows are weakly and the columns are strictly increasing.^{1}

Given a (semistandard) tableau τ we consider the trace weight (cf. [1, 2, 3])

w(τ) = ∏_{(i,j)} x_{j−i}

where the product is over all cells (i, j) of the shape of τ. In order to facilitate notation, we write x(−∞, n) := ∏_{i=−∞}^{n} x_{i}, and x(m, n) := x(−∞, n)/x(−∞, m).

The purpose of this note is to present the following two product formulae for trace generating functions for semistandard tableaux of unusual shapes.

2000 Mathematics Subject Classification. Primary 05A15 Secondary 05A10, 05E10, 11P81, 15A15.

Key words and phrases. Semistandard tableaux of skew shape, trace generating functions.

†This note grew out of a discussion of Sergey Fomin and Richard Stanley. .

1We emphasise that, in this note, the r-tuple μ will not be a partition, so that the tableaux which we are considering here do not not have skew shapes in the classical sense.

Theorem 1. Let λ = (λ_{1}, λ_{2}, …, λ_{r}) be a partition. The trace generating function ∑_{τ} w(τ), where the sum is over all tableaux τ of shape λ/(0, 0, 1, 1, …), is given by

\frac{x_{−1} ∏_{i=1}^{r} x(−i + 1, λ_{i} − i)^{i−1} ∏_{1≤i<j≤r}(1 − x(0, λ_{j} − j, λ_{i} − i))}{∏_{i=1}^{r} ∏_{ℓ=−r+1}^{λ_{i}−i−1}(1 − x(ℓ, λ_{i} − i))}

× \frac{∏_{i=2}^{r−1}(1 − x_{−1} ∏_{j=1}^{r} x(−j, λ_{j} − j)/x(−r, −i))}{∏_{i=1}^{r}(1 − x_{−1} ∏_{j=1}^{r} x(−j, λ_{j} − j)/x(−r, λ_{i} − i))}. (2)

Theorem 2. Let λ = (λ_{1}, λ_{2}, …, λ_{r}) be a partition. The trace generating function ∑_{τ} w(τ), where the sum is over all semistandard tableaux τ of shape λ/(0, 1, 2, 2, 2, …), is given by

\frac{x_{0} ∏_{i=1}^{r} x(−i + 2, λ_{i} − i)^{i−1} ∏_{1≤i<j≤r}(1 − x(0, λ_{j} − j, λ_{i} − i))}{∏_{i=1}^{r} ∏_{ℓ=−r+2}^{λ_{i}−i−1}(1 − x(ℓ, λ_{i} − i))}

× \frac{∏_{i=2}^{r−2}(1 − x_{0} ∏_{j=1}^{r} x(−j + 1, λ_{j} − j)/x(−r + 1, −i))}{∏_{i=1}^{r}(1 − x_{0} ∏_{j=1}^{r} x(−j + 1, λ_{j} − j)/x(−r + 1, λ_{i} − i))}. (3)

Remark. Both expressions should have a nicer formulation when using Gansner’s [2] “trace hook lengths.”

We provide the details for the proof of Theorem 1, the proof of Theorem 2 being similar.

Proof of Theorem 1. From [3, Theorem 7.2], we know that the generating function for tableaux of shape λ/(0, 0, …) with a_{1} being the first entry in the first row and a_{2} being the first entry in rows 2, 3, …, r is given by

det_{1≤i,j≤r} (\frac{x(−1, λ_{i} − i)^{a_{1}}}{∏_{ℓ=0}^{λ_{i}−i−1}(1 − x(ℓ, λ_{i} − i))} ⋮ \frac{x(−j, λ_{i} − i)^{a_{2}}}{∏_{ℓ=−j+1}^{λ_{i}−i−1}(1 − x(ℓ, λ_{i} − i))}).

In the determinant the row index is i and the column index is j. The notation of the determinant means that the first expression gives the first column and the second expression gives the remaining columns, j ≥ 2.

Hence, the desired trace generating function is given by

∑_{0≤a_{1}<a_{2}} x(−r, −2)^{−a_{2}} det_{1≤i,j≤r} (\frac{x(−1, λ_{i} − i)^{a_{1}}}{∏_{ℓ=0}^{λ_{i}−i−1}(1 − x(ℓ, λ_{i} − i))} ⋮ \frac{x(−j, λ_{i} − i)^{a_{2}}}{∏_{ℓ=−j+1}^{λ_{i}−i−1}(1 − x(ℓ, λ_{i} − i))}).

If we sum over a_{1}, then we obtain

∑_{0≤a_{2} (sic!)} x(−r, −2)^{−a_{2}} det_{1≤i,j≤r} (\frac{1 − x(−1, λ_{i} − i)^{a_{2}}}{∏_{ℓ=−1}^{λ_{i}−i−1}(1 − x(ℓ, λ_{i} − i))} ⋮ \frac{x(−j, λ_{i} − i)^{a_{2}}}{∏_{ℓ=−j+1}^{λ_{i}−i−1}(1 − x(ℓ, λ_{i} − i))}).

In the determinant add x_{−1}^{−a_{2}} times the second column to the first column to obtain

∑_{0≤a_{2} (sic!)} x(−r, −2)^{−a_{2}} det_{1≤i,j≤r} (\frac{1}{∏_{ℓ=−1}^{λ_{i}−i−1}(1 − x(ℓ, λ_{i} − i))} ⋮ \frac{x(−j, λ_{i} − i)^{a_{2}}}{∏_{ℓ=−j+1}^{λ_{i}−i−1}(1 − x(ℓ, λ_{i} − i))}).

Now we expand the determinant along the first column,

∑_{k=1}^{r} \frac{(−1)^{k−1}}{∏_{ℓ=−1}^{λ_{k}−k−1}(1 − x(ℓ, λ_{k} − k))}

× ∑_{0≤a_{2}} x(−r, −2)^{−a_{2}} det_{i=1,...,k−1,k+1,...,r}_{j=1,...,r−1}(\frac{x(−j − 1, λ_{i} − i)^{a_{2}}}{∏_{ℓ=−j}^{λ_{i}−i−1}(1 − x(ℓ, λ_{i} − i))}) (4)

Next we compute the determinant in (4). By taking

(\frac{x_{−1} ∏_{i=1}^{r} x(−i, λ_{i} − i)}{x(−r, λ_{k} − k)})^{a_{2}}

out of the determinant in (4), a determinant of the form

det_{1≤i,j≤r−1} (\frac{1}{∏_{ℓ=−j}^{μ_{i}−1}(1 − x(ℓ, μ_{i}))})  (5)

remains. This determinant equals

\frac{∏_{i=1}^{r−1} ∏_{ℓ=−r+1}^{−i−1} x(ℓ, 0)}{∏_{i=1}^{r−1} ∏_{ℓ=−r+1}^{μ_{i}−1}(1 − x(ℓ, μ_{i}))} det_{1≤i,j≤r−1} ((x(0, −r + 1) − x(0, μ_{i})) ⋯ (x(0, −j − 1) − x(0, μ_{i}))).

The last determinant can be reduced to a Vandermonde determinant by elementary column operations, hence the determinant in (5) equals

\frac{∏_{ℓ=−r+1}^{−2} x(ℓ, 0)^{−ℓ−1}}{∏_{i=1}^{r−1} ∏_{ℓ=−r+1}^{μ_{i}−1}(1 − x(ℓ, μ_{i}))} × ∏_{1≤i,j≤r−1}(x(0, μ_{j}) − x(0, μ_{i})).  (6)

If in (4) we use the equality of (5) and (6) and moreover sum over a_{2}, the expression (4) reduces to

∑_{k=1}^{r} \frac{(−1)^{k−1}}{∏_{ℓ=−1}^{λ_{k}−k−1}(1 − x(ℓ, λ_{k} − k))} \frac{1}{(1 − x_{−1} ∏_{i=1}^{r} x(−i, λ_{i} − i)/x(−r, λ_{k} − k))}

× \frac{∏_{ℓ=−r+1}^{−2} x(ℓ, 0)^{−ℓ−1}}{_{∏_{i=1}^{r} ; i≠k} ∏_{ℓ=−r+1}^{λ_{i}−i−1}(1 − x(ℓ, λ_{i} − i))}

× \frac{∏_{1≤i<j≤r}(x(0, λ_{j} − j) − x(0, λ_{i} − i))}{∏_{i=1}^{j−1}(x(0, λ_{j} − j) − x(0, λ_{i} − i)) ∏_{i=j+1}^{r}(x(0, λ_{i} − i) − x(0, λ_{j} − j))},

which equals

\frac{∏_{ℓ=−r+1}^{−2} x(ℓ, 0)^{−ℓ−1} ∏_{1≤i<j≤r}(x(0, λ_{j} − j) − x(0, λ_{i} − i))}{∏_{i=1}^{r} ∏_{ℓ=−r+1}^{λ_{i}−i−1}(1 − x(ℓ, λ_{i} − i))}

∑_{k=1}^{r} \frac{∏_{i=2}^{r−1}(1 − x(−i, λ_{k} − k))}{_{∏_{i=1}^{r} ; i≠k}(x(0, λ_{i} − i) − x(0, λ_{j} − j))} \frac{1}{(1 − x_{−1} ∏_{i=1}^{r} x(−i, λ_{i} − i)/x(−r, λ_{k} − k))}. (7)

To evaluate the sum we apply the partial fraction expansion

\frac{z p(z)}{∏_{i=1}^{r}(y_{i} − z)} = ∑_{k=1}^{r} \frac{p(y_{k})}{_{∏_{i=1}^{r} ; i≠k}(y_{i} − y_{k})} \frac{1}{1 − z/y_{k}} ,

(here, p must be a polynomial of degree at most r − 2), with p(z) = ∏_{i=2}^{r−1}(1 − x(−i, 0)z), y_{i} = x(0, λ_{i} − i), and z = x_{−1} ∏_{i=1}^{r} x(−i, λ_{i} − i)/x(−r, 0). Application of this partial fraction expansion in (7) and some manipulation finally lead to the expression in (2). □

References

1. E. Gansner, The enumeration of plane partitions via the Burge correspondence, Illinois J. Math. 25 (1981), 533–554.

2. E. Gansner, The Hillman–Grassl correspondence and the enumeration of reverse plane partitions, J. Combin. Theory A 30 (1981), 71–89.

3. C. Krattenthaler, Generating functions for plane partitions of a given shape, Manuscripta Math. 69 (1990), 173–202.

Fakultät für Mathematik der Universität Wien, Nordbergstrasze 15, A-1090 Wien, Austria.