By Mariolys Rivas

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**Extra resources for D-finite symmetric functions**

**Example text**

2. There exists another argument based on singularity analysis to prove that a function is not D-ﬁnite. 3 (Folklore). The number of partitions of n, denoted by p(n), is not P-recursive. In order to prove this, it suﬃces to show that the ordinary generating series ∑ ∏ 1 p(n)tn = 1 − tk n≥0 k is not D-ﬁnite. For this, notice that such generating function has an inﬁnite number of singularities, which is not possible for a D-ﬁnite power series. We do not cover this type of analysis in our work, the reader is directed to Flajolet and Sedgewick [6] for further details.

Recall that two bases {uλ } and {vλ } are said to be adjoint if ⟨uλ , vµ ⟩ = δλµ for all partitions λ and µ. CHAPTER 6. 5. 5, ( Dpλ (pk )Dpµ (pk ) λ,µ = h(t) ∑ Dpλ (pk )Dpλ (pk ) λ ( pλ ∗ pµ zλ zµ ) pλ zλ = h(t) Dp(0) (pk )Dp(0) (pk )p(0) + Dpk (pk )Dpk (pk ) pk ) k = h(t)(p2k + kpk ). Denote by Λ∗ the algebra of symmetric power series. The map ⟨⟨⟩⟩ : Λ → Λ∗ given by ∑ ⟨⟨sλ ⟩⟩ = n sλ[n] , is called the reduced notation for sλ , which can be extended by linearity, since the Schur functions form a basis of Λ.

Then, Kµ1 the number of standard Young tableaux of shape µ is equal to n Kµ1 = ∏ n! , x∈T hook(x) n where Kµ1 is the Kostka number of shape µ and type 1n . This formula has been proven in many diﬀerent ways, from both algebraic and combinatorial points of view. We direct the reader to the recent proof [2] by Bandlow. 3. Denote by am the number of standard Young tableaux of shape (m−|λ|, λ), for a ﬁxed partition λ. Then {am }m≥0 is P-recursive. Proof. Using the hook formula we get: am = = ∏ m! x∈T(m−|λ|,λ) hook(x) m!