Direct sum of permutations

In combinatorics, the skew sum and direct sum of permutations are two operations to combine shorter permutations into longer ones. Given a permutation π of length m and the permutation σ of length n, the skew sum of π and σ is the permutation of length m + n defined by

and the direct sum of π and σ is the permutation of length m + n defined by

The skew sum of the permutations π = 2413 and σ = 35142 is 796835142 (the last five entries are equal to σ, while the first four entries come from shifting the entries of π) while their direct sum is 241379586 (the first four entries are equal to π, while the last five come from shifting the entries of σ).

If M_π and M_σ are the permutation matrices corresponding to π and σ, respectively, then the permutation matrix ${\displaystyle M_{\pi \ominus \sigma }}$ corresponding to the skew sum ${\displaystyle \pi \ominus \sigma }$ is given by

and the permutation matrix ${\displaystyle M_{\pi \oplus \sigma }}$ corresponding to the direct sum ${\displaystyle \pi \oplus \sigma }$ is given by

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