Symmetric groups and free probability

In Chapter 7, we showed that the symbols (Σ_k ) _k≥1 and (p_k ) _k≥1 form two algebraic bases of the algebra of observables O https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315371016/5e644d52-ef68-4f0e-ac73-bbf480af375a/content/eq5710.tif"/> , and that the restriction on O https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315371016/5e644d52-ef68-4f0e-ac73-bbf480af375a/content/eq5711.tif"/> of the natural gradation deg(·) of the algebra of partial permutations ℂ P S ( ∞ ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315371016/5e644d52-ef68-4f0e-ac73-bbf480af375a/content/eq5712.tif"/> is such that deg   Σ k   =   deg   p k   =   k and deg ( Σ k   −   p k )   ≤   k   −   1 ; https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315371016/5e644d52-ef68-4f0e-ac73-bbf480af375a/content/eq5713.tif"/>