Noncommutative symmetric functions and Lagrange inversion. (English) Zbl 1133.05101
Summary: We compute the non-commutative Frobenius characteristic of the natural action of the 0-Hecke algebra on parking functions, and obtain as corollaries various forms of the non-commutative Lagrange inversion formula.
MSC:
| 05E05 | Symmetric functions and generalizations |
| 20C08 | Hecke algebras and their representations |
| 05A15 | Exact enumeration problems, generating functions |
| 16W30 | Hopf algebras (associative rings and algebras) (MSC2000) |
Keywords:
Lagrange inversion; parking functions; noncommutative symmetric functions; 0-Hecke algebrasOnline Encyclopedia of Integer Sequences:
Motzkin numbers: number of ways of drawing any number of nonintersecting chords joining n (labeled) points on a circle.Catalan’s triangle T(n,k) (read by rows): each term is the sum of the entries above and to the left, i.e., T(n,k) = Sum_{j=0..k} T(n-1,j).
Catalan triangle A009766 transposed.
Triangle read by rows: T(n, k) = [x^k] x*Pochhammer(n + x, n)/(n + x).
a(n) = binomial(4*n+1,n)*2/(3*n+2).
Triangle read by rows: T(n,k) is the number of dissections of a convex n-gon by nonintersecting diagonals, having a k-gon over a fixed edge (base).
A triangle of Motzkin ballot numbers.
Riordan array (f(x)^3, f(x)), where 1 + x*f^3(x)/(1 - x*f(x)) = f(x).
Riordan array (f(x)^4, f(x)), where 1 + x*f^4(x)/(1 - x*f(x)) = f(x).
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