Operad: Poisson

abbreviations

Pois, Poiss

category

Zvec

type

symetrique

dimensions

1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800

OEIS: A000142

Formula: factorial(n)

Serie: x/(1-x)

Actions: s[1], t*s[1, 1] + s[2], t*s[1, 1, 1] + (t^2+t)*s[2, 1] + s[3], t^2*s[1, 1, 1, 1] + (t^3+t^2+t)*s[2, 1, 1] + 2*t^2*s[2, 2] + (t^3+t^2+t)*s[3, 1] + s[4], t^2*s[1, 1, 1, 1, 1] + (t^4+2*t^3+t^2)*s[2, 1, 1, 1] + (t^4+2*t^3+2*t^2)*s[2, 2, 1] + (t^4+3*t^3+t^2+t)*s[3, 1, 1] + (t^4+2*t^3+2*t^2)*s[3, 2] + (t^4+t^3+t^2+t)*s[4, 1] + s[5], t^3*s[1, 1, 1, 1, 1, 1] + (t^5+2*t^4+t^3+t^2)*s[2, 1, 1, 1, 1] + (2*t^5+3*t^4+4*t^3)*s[2, 2, 1, 1] + (3*t^4+t^3+t^2)*s[2, 2, 2] + (t^5+5*t^4+3*t^3+t^2)*s[3, 1, 1, 1] + (3*t^5+6*t^4+5*t^3+2*t^2)*s[3, 2, 1] + (t^5+t^4+3*t^3)*s[3, 3] + (2*t^5+3*t^4+3*t^3+t^2+t)*s[4, 1, 1] + (t^5+4*t^4+2*t^3+2*t^2)*s[4, 2] + (t^5+t^4+t^3+t^2+t)*s[5, 1] + s[6]

properties

Koszul, binary, quadratic, cyclic

Koszul dual operad

Pois

generateurs

• generator * of type s[2] of degree 0
• generator c of type s[1, 1] of degree 1

relations

• equality: (x*y)*z = x*(y*z)
• equality: xc(y*z) = (xcy)*z+y*(xcz)
• relation: (xcy)cz+(ycz)cx+(zcx)cy

morphisms

→ Com, → Lie, → Ram

References

• B. Fresse, 2006 ↗