A bijective proof of Lassalle’s partition identity

Jiang Zeng

Institut Girard Désargues, Université Claude Bernard Lyon 1 43, boulevard du 11 Novembre 1918, F-69622 Villeurbanne Cedex e-mail: zeng@desargues.univ-lyon1.fr

October 15, 2002

We give a bijective proof of the following form of a conjecture in [2, 3]. The special s = 1 case was proved in the same vein by Simion [2]. For other proofs of this conjecture, see [4]. Our notations are the same as in [2, 3]. For an integer partition μ let the length l(μ) be the number of the parts of μ, m_{i}(μ) the number of parts equal to i, z_{μ} = ∏_{i≥1} i^{m_{i}(μ)}m_{i}(μ)! and ⟨{μ \atop r}⟩ the number of ways to choose r different cells from the diagram of the partition μ taking at least one cell from each row. For any positive integer n let (a)_{n} = a(a + 1) ⋯ (a + n − 1) and [n] = {1, 2, …, n}.

Theorem 1 We have

∑_{|μ|=n} \frac{n!}{z_{μ}} ⟨{μ \atop r}⟩X^{l(μ)−1} ∑_{i=1}^{l(μ)} μ_{i}\binom{μ_{i} + s − 1}{s − 1}

= \binom{n}{r}(s + r)_{n−r} [(X + s)_{r} − (X)_{r}] .  (1)

Recall that the functional graph of an injection f : [n] → [n + r] consists of disjoint cycles and paths. Each path is of form a_{0} → a_{1} → ⋯ → a_{l}, where f(a_{i}) = a_{i+1} for 0 ≤ j < l, with f^{−1}(a_{0}) empty, and a_{l} ∈ [n + r] \ [n]. Each cycle is of the form a_{0} → a_{1} → ⋯ → a_{l}, where f(a_{i}) = a_{i+1} for 0 ≤ j ≤ l, with a_{l+1} = a_{0}. Let cyc f be the number of cycles in f. Then the corresponding generating function reads as follows:

∑_{f:[n]→[n+s]} X^{cyc f} = (X + s)_{n}  (2)

where f is an injection. The above identity seems to be first published by Foata and Strehl [1] and can be readily verified by induction on n as in the well-known case of permutations, which corresponds to s = 0.

We shall prove that the two sides of (1) are the generating function of pairs (f, R) with respect to the number of cycles of f, where f is an injection from [n] to [n + s] and R is a r-subset of [n] such that each cycle of f has at least one element in R and the restriction of f on R is not a permutation of R.

Indeed, to construct such a pair (f, R) we can first choose r elements of [n] to form a r-subset R, there are \binom{n}{r} ways to do so; and construct an injection g : R → R ∪ {n + 1, …, n + s} such that its restriction on R is not a permutation of R, the generating function of such injections is then (X +s)_{r} −(X)_{r}; finally insert succesively the n−r elements of [n]\R into the above injection g to obtain the injection f : [n] → [n+s], there are (s+r)_{n−r} ways to insert the n − r elements into any above injection g. This gives the right-hand side of (1).

On the other hand, we can construct such a pair (f, R) as follows: construct first a permutation σ on [n], for any partition μ of n there are n!/z_{μ} such permutations of type μ; choose r elements of [n] to form the subset R such that each cycle of σ has at least one element in R, there are ⟨{μ \atop r}⟩ ways to do so; choose any cycle (a, σ(a), ⋯, σ^{μ_{i}−1}(a)) of σ with length μ_{i} to form s paths:

a → σ(a) → ⋯ σ^{l_{1}−1}(a) → n + 1,
σ^{l_{1}}(a) → σ^{l_{1}+1}(a) → ⋯ σ^{l_{2}−1}(a) → n + 2,
· · · · · · → ⋯
σ^{l_{s−1}}(a) → σ^{l_{s−1}+1}(a) → ⋯ σ^{l_{s}−1}(a) → n + s,

where l_{1} + l_{2} + ⋯ l_{s} = μ_{i} and by convention the corresponding path is empty if l_{i} = 0. Obviously there are μ_{i}\binom{μ_{i}+s−1}{s−1} ways to form such s paths starting from any cycle of length μ_{i}. Finally we complete the construction of f : [n] → [n + s] by using the s paths above and the remaining l(μ) − 1 cycles of σ. This gives the left-hand side of (1).

The theorem follows then from the two above countings.

References

[1] Foata (D.) and Strehl (V.) Combinatorics of Laguerre polynomials, in : Enumeration and Design, Waterloo Jubilee Conference (Academic Press, New York, 1984) 123–140.

[2] Lassalle (M.) Quelques conjectures combinatoires relatives à la formule classique de Chu-Vandermonde, Adv. in Appl. Math. 21 (1998), 457-472.

[3] Lassalle (M.) Une conjecture en théorie des partitions, http://xxx.lanl.gov/math.CO/9901040, 1999.

[4] Lassalle (M.) Une identité en théorie des partitions, Journal of Combinatorial Theory, Series A, to appear.