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A property of dominance of partitions
Mireille Bousquet-Mélou
To cite this version:
Mireille Bousquet-Mélou. A property of dominance of partitions. 2008. �hal-00265335�
hal-00265335, version 1 - 18 Mar 2008
A PROPERTY OF DOMINANCE ON PARTITIONS
MIREILLE BOUSQUET-M´ELOU
Abstract. Given an integer partitionλ= (λ1, . . . , λℓ) and an integerk, denote byλ(k) the sequence of lengthℓobtained by reordering the values|λi−k|in non-increasing order. Ifλ dominatesµand has the same weight, thenλ(k)dominatesµ(k).
Apartitionof an integerninkparts is a non-increasing sequenceλ= (λ1, . . . , λk) of positive integers that sum ton. The integernis called theweightofλ, denoted||λ||, whilekis thelength ofλ, denotedℓ(λ). Partitions are partially ordered by dominance:
λ≥µ if λ1+· · ·+λi ≥µ1+· · ·+µi for alli,
where it is understood that λi = 0 if i > ℓ(λ) (and similarly for µ). Note that λ≥µ implies in particular that ||λ|| ≥ ||µ||. The definition of dominance can be extended to finite non- increasing sequences of non-negative integers. Observe that adding some zeroes at the end of two such sequencesλand µdoes not affect their dominance relation.
Definition 1. For an ℓ-tuple λ= (λ1, . . . , λℓ) of non-negative integers and k ∈N, let λ(k) be the sequence of lengthℓ obtained by reordering the values|λi−k|in non-increasing order.
For instance, ifλ= (4,2,1,0), then
λ(1) = (3,1,1,0), λ(2)= (2,2,1,0), λ(3)= (3,2,1,1), λ(4)= (4,3,2,0).
The following result was observed experimentally by Jean Creignou in his work with Herv´e Diet on codes in unitary groups and Schur polynomials. More precisely, it was useful when implementing their technique to obtain certain bounds on the size of codes [2].
Proposition 2. Let λ= (λ1, . . . , λℓ)andµ= (µ1, . . . , µℓ)be two partitions of the same weight, possibly completed with zeroes so that they have the same length. Assume that λdominates µ.
Then for all positive integerk,λ(k) dominates µ(k).
Example. Takeλ= (4,2,1,0) as above andµ= (4,1,1,1). These two partitions have the same weight, andλ≥µ. The sequencesλ(k) are listed above for 1≤k≤4, and
µ(1)= (3,0,0,0), µ(2)= (2,1,1,1), µ(3)= (2,2,2,1), µ(4)= (3,3,3,0).
It is easy to check that the proposition holds on this example.
Observe that the proposition would be obvious fornegativeintegersk=−m: in the construc- tion ofλ(k), every part ofλis simply increased bym, and the order of the parts does not change.
For similar reasons, the proposition is clear whenk≥λ1: the valuesλi−kandµi−kare non- positive for alli, so that the order of the parts is simply reversed: λ(k)= (k−λℓ, . . . , k−λ2, k−λ1).
The fact that reversion is involved also shows that the result cannot hold for partitions of different weights. Take for instanceλ= (2),µ= (1) andk= 2.
Proof. Recall that λcovers µif there exist no ν such that λ > ν > µ. The covering relations for the dominance order on partitions of the same weight were described by Brylawski [1]. The partitionλ= (λ1, λ2, . . .) covers the partition µ= (µ1, µ2, . . .) iff there existsi < j such that
λi=µi+ 1, λj=µj−1, λp=µp forp6∈ {i, j}, and eitherj =i+ 1 orµi=µj. Let us prove the proposition whenλcoversµ. The general case then follows by transitivity.
Recall that λ(k) is obtained by reordering the multiset Mλ(k) ={|λp−k|,1 ≤p≤ ℓ}. Denote
Date: March 18, 2008.
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2 M. BOUSQUET-M ´ELOU
similarly Mµ(k)={|µp−k|,1 ≤p≤ℓ}. Then Mλ(k) is obtained fromMµ(k) by replacing a copy of|µi−k|by|µi+ 1−k|, and a copy of|µj−k|by|µj−1−k|. We study separately 5 cases, depending on the value ofk. For each of them, we describe howλ(k)is obtained fromµ(k). From this description, it should be clear that the dominance relation is preserved.
• Ifk < µj, the first occurrence ofµi−k inµ(k)is replaced by µi−k+ 1, while the last occurrence ofµj−kis replaced byµj−k−1.
• Ifk=µj< µi, the first occurrence ofµi−kinµ(k) is replaced byµi−k+ 1, while the first occurrence of 0 is replaced by 1.
• Ifk=µj =µi, the first two copies of 0 inµ(k) are both replaced by 1.
• Ifµj < k ≤µi, thenµ(k) contains entriesµi−k andk−µj, in some order. To obtain λ(k), the first occurrence ofµi−kis replaced byµi−k+ 1, and the first occurrence of k−µj is replaced byk−µj+ 1.
• Ifk > µi , then the valuesk−µj occur before the valuesk−µi inµ(k). To obtainλ(k), the first occurrence ofk−µj is replaced byk−µj+ 1, and the first occurrence ofk−µi is replaced byk−µi−1.
References
[1] Thomas Brylawski. The lattice of integer partitions.Discrete Math.6(1973) 201–219.
[2] Jean Creignou and Herv´e Diet, Linear programming bounds for unitary space time codes, Arxiv:0803.1227, 2008.
M. Bousquet-M´elou: CNRS, LaBRI, Universit´e Bordeaux 1, 351 cours de la Lib´eration, 33405 Talence, France
E-mail address: mireille.bousquet@labri.fr
