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Multi-indices and ordered monomials

Lax825748.Monomials · concepts/Lax825748/Monomials.lean · lax-825748

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    Natural Language Statement

    Definition

    For a multi-index I=(i1,…,id)I=(i_1,\ldots,i_d), write ∣I∣=∑jij|I|=\sum_j i_j and I!=∏jij!I!=\prod_j i_j!. For an ordered family of noncommutative polynomials, PI=P1i1⋯PdidP^I=P_1^{i_1}\cdots P_d^{i_d} uses that order of multiplication. The empty family has total degree zero, factorial one, and monomial one.

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    DefinitionThis conceptRelated conceptA → B: B builds on ADescendants are omitted for concepts with more than 10 descendants.

    Lean source view on GitHub

    1import Lax825748.Series
    2import Mathlib.Data.Nat.Factorial.Basic
    3import Mathlib.Algebra.BigOperators.Group.Finset.Basic
    4
    5/-!
    6---
    7title: Multi-indices and ordered monomials
    8type: definition
    9---
    10For a multi-index I=(i1,…,id)I=(i_1,\ldots,i_d), write ∣I∣=∑jij|I|=\sum_j i_j and
    11I!=∏jij!I!=\prod_j i_j!. For an ordered family of noncommutative polynomials,
    12PI=P1i1⋯PdidP^I=P_1^{i_1}\cdots P_d^{i_d} uses that order of multiplication.
    13The empty family has total degree zero, factorial one, and monomial one.
    14-/
    15
    16noncomputable section
    17
    18namespace Lax825748.Monomials
    19
    20open Series
    21open scoped BigOperators
    22
    23def totalDegree {d : ℕ} (I : Fin d → ℕ) : ℕ := ∑ j, I j
    24
    25def multiFactorial {d : ℕ} (I : Fin d → ℕ) : ℕ := ∏ j, (I j).factorial
    26
    27noncomputable def orderedPowers {X : Type} {d : ℕ}
    28 (P : Fin d → NCPolynomial X) (I : Fin d → ℕ) : NCPolynomial X :=
    29 (List.ofFn fun j => P j ^ I j).prod
    30
    31end Lax825748.Monomials
    32
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