Séminaire Lotharingien de Combinatoire 47 (2002), Article B47d

HOOKS AND POWERS OF PARTS IN PARTITIONS

ROLAND BACHER AND LAURENT MANIVEL

Abstract. This paper shows that the number of hooks of length k contained in all partitions of n equals k times the number of parts of length k in partitions of n. It contains also formulas for the moments (under uniform distribution) of k-th parts in partitions of n.

1. Introduction and Main Results

Many textbooks contain material on partitions. Two standard references are [A] and [S].

A partition of a natural integer n with parts λ_{1}, …, λ_{k} is a finite decreasing sequence λ = (λ_{1} ≥ λ_{2} ≥ ⋯ ≥ λ_{k} > 0) of natural integers λ_{1}, …, λ_{k} > 0 such that n = ∑_{i=1}^{k} λ_{i}. We denote by |λ| the content n of λ. Partitions are also written as sums: n = λ_{1} + ⋯ + λ_{k} and one uses also the (abusive) multiplicative notation

λ = 1^{ν_{1}} · 2^{ν_{2}} ⋯ n^{ν_{n}}

where ν_{i} denotes the number of parts equal to i in the partition λ.

A partition is graphically represented by its Young diagram obtained by drawing λ_{1} adjacent boxes of identical size on a first row, followed by λ_{2} adjacent boxes of identical size on a second row and so on with all first boxes (of different rows) aligned along a common first column. In the sequel we identify a partition with its Young diagram. A hook in a partition is a choice of a box H in the corresponding Young diagram together with all boxes at the right of the same row and all boxes below of the same column. The total number of boxes in a hook is its hooklength, the number of boxes in a hook to the right of H is its armlength and the number of boxes of a hook below H is the leglength. The Figure below displays the Young diagram of the partition (5, 4, 3, 1) of 13 together with a hook of length 4 having armlength 2 and leglength 1. We call the couple (armlength,leglength) of a hook its hooktype and denote it by τ = τ(α, k − 1 − α) if its armlength is α and its leglength k − 1 − α. Such a hook has hence total length k and there are exactly k different hooktypes for hooks of length k.

2000 Mathematics Subject Classification. 05A17.

Key words and phrases. Partition.

The partition (5, 4, 3, 1) of 13 together with a hook of type τ(2, 1) and length 4.

Let k be a natural integer and let τ = τ(α, k−1−α) be the hooktype of a hook of length k with armlength α and leglength k − 1 − α. Given a partition λ of n, set

τ(λ) = ♯{hooks of type τ in (the Young diagram of) λ}

and

τ(n) = ∑_{λ, |λ|=n} τ(λ)

where the sum is over all partitions of n.

Theorem 1.1. One has

\begin{aligned}∑_{n=1}^{∞} τ(n)z^{n} & = & \frac{z^{k}}{1 − z^{k}} ∏_{i=1}^{∞} \frac{1}{1 − z^{i}} \\ & = & ∑_{λ=1^{ν_{1}}2^{ν_{2}}···} ν_{k} z^{|λ|} \end{aligned}

where the last sum is over all partitions of integers.

In other terms, the number of hooks of given type and length k appearing in all partitions of n equals the number of parts of length k in all partitions of n.

This result implies in particular that the total number of hooks of given type τ = τ(α, k − 1 − α) occuring in all partitions of n depends only on the length k and not on the particular hooktype τ(α, k −1−α) itself. Since there are exactly k distinct hooktypes for hooks of length k, the total number of hooks of length k in partitions of n is given by the coefficient of z^{n} of the series

k \frac{z^{k}}{1 − z^{k}} ∏_{i=1}^{∞} \frac{1}{1 − z^{i}} .

C. Bessenrodt pointed out to us that this Theorem follows directly from Theorem 1.1 in [B]. Indeed, Theorem 1.1 of [B] states that the number of k-hooks of given leglength which can be added to a Young diagram always exceeds by 1 the number of k-hooks of the same leglength which can be removed from the same Young diagram. This implies the identity

∑_{n=1}^{∞} τ(n)z^{n} = z^{k} (∑_{n=1}^{∞} τ(n)z^{n} + ∏_{i=1}^{∞} \frac{1}{1 − z^{i}} )

on the generating series appearing in Theorem 1.1. The obvious initial conditions τ(n) = 0 for n < k determine now the generating series which is easily checked to be

\frac{z^{k}}{1 − z^{k}} ∏_{i=1}^{∞} \frac{1}{1 − z^{i}} .

Our initial proof of Theorem 1.1 was based on the q−binomial theorem.

As remarked previously, partitions of a natural integer n can be written in (at least) two different ways: either by considering the finite decreasing sequence

λ = (λ_{1} ≥ λ_{2} ≥ ⋯ ≥ λ_{k})

of its parts or by considering the vector

ν = (ν_{1}, ν_{2}, …, ν_{n})

where ν_{i} counts the multiplicity of parts with length i in λ. We still denote by λ the vector

(λ_{1}, λ_{2}, … λ_{k}, 0, …, 0) ∈ Z^{n}

of length n obtained by appending n − k zero-coordinates at the end of the vector (λ_{1}, …, λ_{k}) defining a partition λ.

We consider moreover the vector γ = (γ_{1}, …, γ_{n}) where γ_{i} equals the number of coordinates equal to i among ν_{1}, …, ν_{n}. The vector γ of a partition encodes “multiplicities of multiplicities (of parts)” and does no longer encode the partition since for instance the γ-vectors of the two partitions 5 = 3+2 and 5 = 4+1 give both rise to γ = (2, 0, 0, 0, 0).

We introduce also the vectors λ(n) = (λ_{1}(n), λ_{2}(n), …, λ_{n}(n)) = ∑_{|λ|=n} λ ,

ν(n) = (ν_{1}(n), ν_{2}(n), …, ν_{n}(n)) = ∑_{|λ|=n} ν ,
γ(n) = (γ_{1}(n), γ_{2}(n), …, γ_{n}(n)) = ∑_{|λ|=n} γ

of Z^{n} obtained by summing up the vectors λ, ν or γ over all partitions of n. The coordinates of the vector λ(n) are of course related to the mean length (under uniform distribution) of the k-th part in partitions of n. Similarly, coordinates of ν(n) relate to the mean multiplicity of parts equal to k and coordinates of γ(n) measure the mean number of distinct part-lengths appearing with common multiplicity k.

The following Table displays all five partitions of 4 together with the corresponding λ-, ν- and γ-vectors.

Partition (1, 1, 1, 1) (2, 1, 1)  (2, 2)  (3, 1)  (4)
λ-vector (1, 1, 1, 1) (2, 1, 1, 0) (2, 2, 0, 0) (3, 1, 0, 0) (4, 0, 0, 0)
ν-vector (4, 0, 0, 0) (2, 1, 0, 0) (0, 2, 0, 0) (1, 0, 1, 0) (0, 0, 0, 1)
γ-vector (0, 0, 0, 1) (1, 1, 0, 0) (0, 1, 0, 0) (2, 0, 0, 0) (1, 0, 0, 0)

Summing up all λ-, ν- and γ-vectors associated to the five partitions of 4 we get hence

λ(4) = (12, 5, 2, 1),
ν(4) = (7, 3, 1, 1),
γ(4) = (4, 2, 0, 1) .

One has the following result.

Theorem 1.2. For all n ≥ 1 we have λ_{n}(n) = ν_{n}(n) = γ_{n}(n) = 1 and

\begin{matrix}λ_{k}(n) = ν_{k}(n) + λ_{k+1}(n) , \\ ν_{k}(n) = γ_{k}(n) + ν_{k+1}(n) \end{matrix}

for k = 1, …, n − 1.

These equalities can be restated as

λ_{k}(n) = ∑_{i=k}^{n} ν_{i}(n) and ν_{k}(n) = ∑_{i=k}^{n} γ_{i}(n) .

This last equality states for instance that the sum over all partitions of n of the number of distinct parts arising with multiplicity at least k equals the number of parts equal to k in all partitions of n. Our proof of this fact uses generating series. C. Bessenrodt ([B1]) communicated to us a beautiful bijective proof which we reproduce with her permission.

The coordinates of the vectors ν(n) are of course given by the generating series mentioned in Theorem 1.1, i.e., the k-th coordinate ν_{k}(n) of ν(n) equals the coefficient of z^{n} in the generating series

\frac{z^{k}}{1 − z^{k}} ∏_{i=1} \frac{1}{1 − z^{i}} .

The coordinates of the vectors λ(n) and ν(n) are then easily computed using Theorem 1.2. More precisely, one has the following result:

Corollary 1.3. (i) The k-th coordinate of the vector λ(n) is the coefficient of z^{n} in the generating series

∏_{i=1}^{∞} \frac{1}{1 − z^{i}} ∑_{j=k}^{∞} \frac{z^{j}}{1 − z^{j}} .

(ii) The k-th coordinate of the vector γ(n) is the coefficient of z^{n} in the generating series

\frac{(1 − z) z^{k}}{(1 − z^{k})(1 − z^{k+1})} ∏_{i=1}^{∞} \frac{1}{1 − z^{i}} .

Given a partition

λ = (λ_{1}, …, λ_{n}) = (1^{ν_{1}} ⋯ n^{ν_{n}})

and an integer d ≥ 0 we introduce the vectors \binom{λ}{d} and \binom{ν}{d} ∈ Z^{n} by setting

\binom{λ}{d} = (\binom{λ_{1}}{d}, …, \binom{λ_{n}}{d}) and \binom{ν}{d} = (\binom{ν_{1}}{d}, …, \binom{ν_{n}}{d})

and define

\binom{λ(n)}{d} = ∑_{|λ|=n} \binom{λ}{d} , \binom{ν(n)}{d} = ∑_{|1^{ν_{1}} 2^{ν_{2}}···|=n} \binom{ν}{d}

with coordinates

\binom{λ_{k}(n)}{d} = ∑_{|λ|=n} \binom{λ_{k}}{d} , \binom{ν_{k}(n)}{d} = ∑_{|1^{ν_{1}} 2^{ν_{2}}···|=n} \binom{ν_{k}}{d} .

The following example shows the vectors \binom{λ}{1} = λ, \binom{λ}{2}, \binom{λ}{3} and \binom{ν}{1} = ν, \binom{ν}{2}, \binom{ν}{3} associated to all five partitions of 4.

Example:

Partition  (1, 1, 1, 1) (2, 1, 1)  (2, 2)  (3, 1)  (4)
\binom{λ}{1}  = (1, 1, 1, 1) (2, 1, 1, 0) (2, 2, 0, 0) (3, 1, 0, 0) (4, 0, 0, 0)
\binom{λ}{2}  = (0, 0, 0, 0) (1, 0, 0, 0) (1, 1, 0, 0) (3, 0, 0, 0) (6, 0, 0, 0)
\binom{λ}{3}  = (0, 0, 0, 0) (0, 0, 0, 0) (0, 0, 0, 0) (1, 0, 0, 0) (4, 0, 0, 0)
Partition  (1, 1, 1, 1) (2, 1, 1)  (2, 2)  (3, 1)  (4)
\binom{ν}{1}  = (4, 0, 0, 0) (2, 1, 0, 0) (0, 2, 0, 0) (1, 0, 1, 0) (0, 0, 0, 1)
\binom{ν}{2}  = (6, 0, 0, 0) (1, 0, 0, 0) (0, 1, 0, 0) (0, 0, 0, 0) (0, 0, 0, 0)
\binom{ν}{3}  = (4, 0, 0, 0) (0, 0, 0, 0) (0, 0, 0, 0) (0, 0, 0, 0) (0, 0, 0, 0)

We have thus

\begin{matrix}\binom{λ(4)}{1} = (12, 5, 2, 1), & \binom{λ(4)}{2} = (11, 1, 0, 0), & \binom{λ(4)}{3} = (5, 0, 0, 0) \\ \binom{ν(4)}{1} = (7, 3, 1, 1), & \binom{ν(4)}{2} = (7, 1, 0, 0), & \binom{ν(4)}{3} = (4, 0, 0, 0) \end{matrix}

The following probably well-known result allows easy computations of the vectors \binom{λ(n)}{d} and \binom{ν(n)}{d}.

Proposition 1.4. For any natural integer d ≥ 0, the k-th coefficients \binom{λ_{k}(n)}{d}, respectively \binom{ν_{k}(n)}{d} (extended by \binom{0}{d} for k > n) have generating series

∑_{n}^{∞} \binom{λ_{k}(n)}{d} z^{n} = (∏_{j=1}^{k−1} \frac{1}{1 − z^{j}} ) (∑_{i=0}^{∞} \binom{i}{d} z^{ik} (∏_{j=1}^{i} \frac{1}{1 − z^{j}} ))

and

∑_{n}^{∞} \binom{ν_{k}(n)}{d} z^{n} = (\frac{z^{k}}{1 − z^{k}} )^{d} (∏_{j=1}^{∞} \frac{1}{1 − z^{j}} ) .

Remark 1.5. One has

∑_{|λ|=n} λ_{k}^{d} = ∑_{i} i! Stirling_{2}(d, i) \binom{λ_{k}(n)}{i}

and

∑_{|1^{ν_{1}} 2^{ν_{2}}···|=n} ν_{k}^{d} = ∑_{i} i! Stirling_{2}(d, i) \binom{ν_{k}(n)}{i}

where Stirling_{2}(d, i) denote Stirling numbers of the second kind, defined by x^{d} = ∑_{i} Stirling_{2}(d, i) x(x − 1) ⋯ (x − i + 1).

Asymptotics are not so easy to work out from the formula for \binom{λ(n)}{d}. Our last result is an equivalent expression for the above series on which asymptotics are easier to see.

We introduce the generating series σ_{r}(k) defined as

σ_{r}(k) = ∑_{i=k}^{∞} (\frac{z^{i}}{1 − z^{i}} )^{r}

for r ≥ 1 and k ≥ 1 natural integers. We consider the series σ_{r}(k) as beeing graded of degree r and define the homogeneous series S_{d}(k) of degree d by

\begin{aligned}S_{d}(k) & = & ∑_{|(1^{ν_{1}} 2^{ν_{2}}···)|=d} \frac{d!}{(∑_{i} ν_{i})!}\binom{(∑_{i} ν_{i})}{ν_{1} ν_{2} …} ∏_{i=1}^{d} (\frac{σ_{i}(k)}{i})^{ν_{i}} \\ & = & d! ∑_{|1^{ν_{1}}2^{ν_{2}}···t^{ν_{t}}|=d}  ∏_{j=1}^{d} \frac{(σ_{j}(k))^{ν_{j}}}{j^{ν_{j}} ν_{j}!} \end{aligned}

(i.e., the coefficient of the homogeneous “monomial” series σ_{λ}(k) = σ_{λ_{1}}(k) … σ_{λ_{s}}(k) equals the number of elements in the symmetric group on |λ| elements of the conjugacy class with s cycles of length λ_{1}, λ_{2}, …, λ_{s}).

We have then the following result.

Theorem 1.6. For any natural integers d ≥ 1 and k ≥ 1, we have

∑_{n}^{∞} \binom{λ_{k}(n)}{d} z^{n} = \frac{S_{d}(k)}{d!} (∏_{j=1}^{∞} \frac{1}{1 − z^{j}} ) .

The first series S_{i} = S_{i}(k) are given in terms of σ_{j} = σ_{j}(k) as follows

S_{0} = 1,

S_{1} = σ_{1},

S_{2} = σ_{1}^{2} + σ_{2},

S_{3} = σ_{1}^{3} + 3σ_{1}σ_{2} + 2σ_{3}

S_{4} = σ_{1}^{4} + 6σ_{1}^{2}σ_{2} + 3σ_{2}^{2} + 8σ_{1}σ_{3} + 6σ_{4}

S_{5} = σ_{1}^{5} + 10σ_{1}^{3}σ_{2} + 20σ_{1}^{2}σ_{3} + 15σ_{1}σ_{2}^{2} + 30σ_{1}σ_{4} + 20σ_{2}σ_{3} + 24σ_{5}

Let us remark that the analogous statement of Theorem 1.6 for the generating series ∑_{n} \binom{ν_{k}}{d}z^{n} boils down to a trivial identity.

The formulas of Theorem 1.6 ease the computations of asymptotics (in n) for λ_{k}(n) and its moments and allow a rederivation of the results contained in [EL] and [VK]: Indeed, the asymptotics of the coefficients in ∑_{n}^{∞} \binom{λ_{k}(n)}{d}z^{n} are essentially given by the asymptotics of

\frac{σ_{1}^{d}(k)}{d!} (∏_{j=1}^{∞} \frac{1}{1 − z^{j}} )

which can be worked out.

2. Proofs

Proof of Theorem 1.2. The partition 1^{n} yields the unique nonzero contribution to λ_{n}(n) and γ_{n}(n) and this contribution equals 1 in both cases. The partition n consisting of a unique part of length n yields the unique non-zero contribution to ν_{n}(n) and this contribution equals again 1.

Given a partition λ = (λ_{1}, …, λ_{k}) of n, the conjugate partition λ^{t} = (λ_{1}^{t}, …, λ_{k′}^{t}) of λ is defined by

λ_{j}^{t} = ♯{i | λ_{i} ≥ j}

(this corresponds to a reflection of the Young diagramm of λ through the main diagonal y = −x). The difference λ_{k} − λ_{k+1} (where nonexisting parts are considered as parts of length 0) equals hence the number ν_{k}^{t} of parts having length k in the transposed partition λ^{t} = (1^{ν_{1}^{t}} 2^{ν_{2}^{t}} ⋯) of λ. Summing over all partitions of n yields then the recursion relation λ_{k}(n) = ν_{k}(n) + λ_{k+1}(n).

The proof of the equality ν_{k}(n) = γ_{k}(n) + ν_{k+1}(n) uses generating series. Introducing the numbers

\begin{matrix}m_{k}(λ) = ♯\{i | ν_{i} ≥ k\} , \\ m_{k}(n) = ∑_{|λ|=n} m_{k}(λ) \end{matrix}

one has obviously γ_{k}(n) = m_{k}(n) − m_{k+1}(n). We have hence to show the equality m_{k}(n) = ν_{k}(n) for 1 ≤ k < n (the equalities m_{n}(n) = ν_{n}(n) = 1 are easy).

We introduce the generating function

ψ_{k}(y, z) = ∑_{λ} y^{m_{k}(λ)}z^{|λ|} .

One has

\begin{matrix}ψ_{k}(y, z) = ∑_{I⊂\{1,2,3,... \}, ♯(I)<∞} (∏_{i∈I} \frac{yz^{ki}}{1 − z^{i}} ) ( ∏_{1≤j∉I} \frac{1 − z^{kj}}{1 − z^{j}} ) \\ = ∏_{j=1}^{∞} (\frac{yz^{kj}}{1 − z^{j}} + \frac{1 − z^{kj}}{1 − z^{j}} ) = ∏_{j=1}^{∞} (\frac{1}{1 − z^{j}} − (1 − y)\frac{z^{kj}}{1 − z^{j}} ) . \end{matrix}

A small computation yields

\frac{∂ψ_{k}(y, z)}{∂y} = ψ_{k}(y, z) ∑_{j=1}^{∞} \frac{z^{kj}}{1 − (1 − y)z^{kj}}

and we have hence

∑_{n=0}^{∞} m_{k}(n) z^{n} = \frac{∂ψ_{k}}{∂y} (1, z) = (∏_{j=1}^{∞} \frac{1}{1 − z^{j}} ) \frac{z^{k}}{1 − z^{k}} = ∑_{n=0}^{∞} ν_{k}(n) z^{n}

(cf. Theorem 1.1 for the last equality) which finishes the proof. QED

C. Bessenrodt’s bijective Proof of Theorem 1.2. We establish first the equality

ν_{k}(n) = ∑_{i=k}^{n} γ_{i}(n)

which corresponds to the second statement of Theorem 1.2. Define the sets

P_{k}(n) = {(i, λ) | |λ| = n, 1 ≤ i ≤ ν_{k}(λ)}

and

Q_{k}(n) = {(j, λ) | |λ| = n, ν_{j}(λ) ≥ k}

(recall that ν_{i}(λ) counts the number of parts with length i in the partition λ). The cardinality ♯(P_{k}(n)) equals obviously ν_{k}(n) and one has similarly ♯(Q_{k}(n)) = ∑_{i=k}^{n} γ_{i}(n) (recall that γ_{i}(λ) counts the number of distinct partlengths whose parts appear with multiplicity i in λ). An element (i, λ) ∈ P_{k}(n) stems from a partition λ having at least i parts equal to k. Erasing i such parts and adding k parts of length i yields a partition μ of n such that μ contains at least k parts of length i. This shows that (i, μ) ∈ Q_{k}(n) and the application (i, λ) ⟼ (i, μ) is clearly a bijection.

The first statement of Theorem 1.2 can be proven similarly by considering the sets

R_{k} = {(i, λ) | |λ| = n, 1 ≤ i ≤ λ_{k}(λ)}

and

S_{k}(n) = {(j, λ) | |λ| = n, j ≤ ∑_{i=k}^{n} ν_{i}(λ)}

and the bijection R_{k}(n) ⟶ S_{k}(n) defined by (i, λ) ⟼ (i, λ^{t}) (this is more or less the proof given above). QED

Corollary 1.3 results immediately from Theorem 1.2 and from the last equality in Theorem 1.1.

Proof of Proposition 1.4. A partition

λ = (λ_{1} ≥ λ_{2} ≥ ⋯ ≥ λ_{k} = i ≥ λ_{k+1} ≥ …)

with λ_{k} = i of n = ∑_{j=1} λ_{j} can be written as

λ = (i+(λ_{1} −i) ≥ i+(λ_{2} −i) ≥ ⋯ ≥ i+(λ_{k−1} −i) ≥ i ≥ λ_{k+1} ≥ …) .

Such partitions are hence in bijection with pairs of partitions

\begin{matrix}α = (α_{1} = (λ_{1} − i) ≥ α_{2} = (λ_{2} − i) ≥ ⋯ ≥ α_{k−1} = (λ_{k−1} − i) ≥ 0) , \\ ω = (ω_{1} = λ_{k+1} ≥ ω_{2} = λ_{k+2} ≥ …) \end{matrix}

with α having at most k − 1 non-zero parts and ω having all parts ≤ i. The conjugate partition α^{t} of α has hence only parts ≤ k − 1. Such a pair α^{t}, ω of partitions yields hence a unique partition with λ_{k} = i of the integer n = ki + ∑_{j=1}^{k−1} α_{j}^{t} + ∑_{j=k+1} ω_{j} and contributes hence with \binom{i}{d} to the k-th coordinate \binom{λ_{k}(n)}{d} of \binom{λ(n)}{d}. Summing up over i ∈ N yields easily the generating series for \binom{λ_{k}(n)}{d}.

Considering the generating series for \binom{ν_{k}(n)}{d} one has

∑_{n} \binom{ν_{k}(n)}{d}z^{n} = (∑_{j} \binom{j}{d}z^{jk}) ∏_{i≠k} \frac{1}{1 − z^{i}}

and the (easy) equality

∑_{j} \binom{j}{d}Z^{j} = \frac{1}{Z} (\frac{Z}{1 − Z} )^{d+1}

implies the result. QED

Proof of Theorem 1.6. Given a partition λ = (λ_{1}, λ_{2}, …) the definition

λ_{k}^{t} = ♯{i | λ_{i} ≥ k}

for the k-th part of its transposed partition λ^{t} = (λ_{1}^{t}, λ_{2}^{t}, …) shows the equalities

∑_{λ} y^{λ_{k}} z^{|λ|} = ∑_{λ} y^{(λ^{t})_{k}} z^{|λ|} = (∏_{i=1}^{k−1} \frac{1}{1 − z^{i}} ) (∏_{j=k}^{∞} \frac{1}{1 − yz^{j}} ) .

Denote this series by φ_{k}(y, z). An easy computation yields

φ_{k}(y, z) = (∏_{i=1}^{∞} \frac{1}{1 − z^{i}} ) ∏_{j≥k} (1 − (y − 1)\frac{z^{j}}{1 − z^{j}} )^{−1} .

Applying the identity

∏_{i} (1 − x_{i})^{−1} = exp (∑_{l=1}^{∞} \frac{∑_{i} x_{i}^{l}}{l} )

of formal power series to the last factor we get

∏_{j≥k} (1 − (y − 1)\frac{z^{j}}{1 − z^{j}} )^{−1} = exp (∑_{l=1}^{∞} \frac{(y − 1)^{l}}{l} σ_{l}(k))

= ∑_{n=0}^{∞} \frac{(y − 1)^{n}}{n!} ∑_{|(1^{ν_{1}}·2^{ν_{2}}···)|=n} \frac{n!}{(∑_{i} ν_{i})!}\binom{∑_{i} ν_{i}}{ν_{1} ν_{2} …} ∏_{i} (\frac{σ_{i}(k)}{i})^{ν_{i}}

= ∑_{n=0}^{∞} \frac{(y − 1)^{n}}{n!} S_{n}(k) .

We have thus

d! ∑_{λ} \binom{λ_{k}}{d}z^{|λ|} = \frac{∂^{d}φ_{k}}{∂y^{d}} (1, z) = (∏_{i=1}^{∞} \frac{1}{1 − z^{i}} ) S_{d}(k)

which finishes the proof by comparison with Proposition 1.4. QED

We thank S. Attal, M-L. Chabanol and C. Krattenthaler for comments and for their interest in this work. We also thank C. Bessenrodt for comments on a preliminary version.

References

[A] G.E. Andrews, The Theory of Partitions, Encyclopedia of Mathematics and its Applications, Addison–Wesley (1976).

[B] C. Bessenrodt, On hooks of Young diagrams, Ann. of Comb. 2, 103– 110 (1998).

[B1] C. Bessenrodt, Electronic letter to the authors, 18 sept. 2001.

[EL] P. Erdös, J. Lehner, The distribution of the number of summands in the partition of a positive integer, Duke Math. Journal 8, 335–345 (1941).

[GR] G. Gasper, M. Rahman, Basic hypergeometric series, Cambridge University Press (Encycl. of Math. and its Appl.), 1990.

[S] Stanley, Enumerative Combinatorics 2, Cambridge University Press (1999).

[ST] M. Szalay, P. Turán, On some problems of the statistical theory of partitions with application to characters of the symmetric group I, Acta Math. Sci. Hung. 29, 361–379 (1977).

[VK] A. Vershik, S. Kerov, Asymptotics of the Plancherel measure of the symmetric group and the limit form of Young tableaux, Soviet Math. Dokl. 18, 1977, 527–531.

Roland Bacher INSTITUT FOURIER Laboratoire de Mathématiques UMR 5582 (UJF–CNRS) BP 74 38402 St MARTIN D’HÈRES Cedex (France) e-mail: Roland.Bacher@ujf-grenoble.fr Laurent Manivel INSTITUT FOURIER Laboratoire de Mathématiques UMR 5582 (UJF–CNRS) BP 74 38402 St MARTIN D’HÈRES Cedex (France) e-mail: Laurent.Manivel@ujf-grenoble.fr