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3 A little backgournd added. Relation with the Johnson scheme made more explicit.

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Some background on Association Schemes.

An association scheme is a set of symmetric boolean matrices $A_1, \dots , A_S$ such that1) $\sum_{i=1}^s A_i =J$ the all-one-matrix2) $A_1 = I$ the identity matrix 3) $\forall i,j \; A_iA_j \in {\rm span} ( A_i )$

The matrices $A_i$ can be seen as adjency matrix for some graph (but I don't think it might help here)

The span{$A_i$} defines an algebra called the Bose-Mesner Algebra. Condition (3) implies that all matrices commute so they diagonalize in the same basis.

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In the case I'm considering here, the dimension of the $A_i$ is ${M \choose N}N!\times {M \choose N}N!$. The $A_i$ are not explicitly defined but we know that $[A_i]_{fg}=[A_i]_{f'g'}$ if there is a permutation $\pi\in S_M$ and a permutation $\tau \in S_N$ such that $g = \pi \circ f \circ \tau$ and $g' = \pi \circ f' \circ \tau$.

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About the Johnson scheme: The $A_i$ have size ${M \choose N}$. The rows and the columns of the matrices are labeled by subsets of size $N$ of {$1,\dots,M$}. (in my case, the labels are injective functions, ie. ordered sets of subsets of size $N$ of {$1,\dots,M$}.

$[A_i]_{ab}=[A_i]_{a'b'}$ if there is a permutation $\pi\in S_M$ such that $\pi(a) = a'$ and $\pi(b) = b'$. (where $\pi(a)$ denotes the subset of {$1,\dots,M$} obtained by applying the permutation $\pi$ to the elements of the sets $a$.

2 fixed typos

This problem arises while studying the complexity of algorithms and I am quite unfamiliar with the subject.

Consider the set F of injective functions from {1..N} to {1..M}

we can define an association scheme on F x F by (f,f') and (g,g') are in the same class if there is a permutation $\pi\in S_M$ and a permutation $\tau \in S_N$ such that $g = \pi \circ f \circ \tau$ and $g' = \pi \circ f' \circ \tau$tau$. I checked that this really defines an association scheme. In a way it is an "ordered" version of the Johnson scheme. It seems to me that it is a natural extension of the Johnson scheme, but I did not find any reference about it. Q1: Has this association scheme ever been studied? What is its name? Q2: Can this scheme be obtained by a combination (tensor product? Suprema?suprema?) of the Johnson scheme and another quantity? More precisely, I am interested in the "Bose-Mesner Algebra" point of view on this scheme. It is known that all the matrices in the algebra defined by this association scheme diagonalize in the same basis. Q3: How can we construct construct/characterize these eigenspaces? 1 # Association scheme on injective functions This problem arises while studying the complexity of algorithms and I am quite unfamiliar with the subject. Consider the set F of injective functions from {1..N} to {1..M} we can define an association scheme on F x F by (f,f') and (g,g') are in the same class if there is a permutation$\pi\in S_M$and a permutation$\tau \in S_N$such that$g = \pi \circ f \circ \tau$and$g' = \pi \circ f' \circ \tau\$

I checked that this really defines an association scheme. In a way it is an "ordered" version of the Johnson scheme. It seems to me that it is a natural extension of the Johnson scheme, but I did not find any reference about it.

Q1: Has this association scheme ever been studied? What is its name?

Q2: Can this scheme be obtained by a combination (tensor product? Suprema?) of the Johnson scheme and another quantity?

More precisely, I am interested in the "Bose-Mesner Algebra" point of view on this scheme. It is known that all the matrices in the algebra defined by this association scheme diagonalize in the same basis.

Q3: How can we construct these eigenspaces?