Hopf cocycle in terms of the Monomial bases for the smash product decomposition of SSym evaluated on the primitive elements of QSym

Cocycle of total degree 0
sigma(M(), M())     =     M0
Cocycle of total degree 2
sigma(M(1), M(1))    =     0.
Cocycles of total degree 3
sigma(M(1), M(2))    =     2M213.
sigma(M(2), M(1))    =     2M132.
Cocycles of total degree 4
sigma(M(1), M(3))    =     2M2134 + M2314 + M3124.
sigma(M(2), M(2))    =     2M1324 + M1342 + M1423 + M2314 + M3124 + M2413 + M3142.
sigma(M(3), M(1))    =     2M1243 + M1342 + M1423.
Cocycles of total degree 5
sigma(M(1), M(4))    =     2M21345 + M23145 + M31245 + M23415 + M41235.
sigma(M(2), M(3))    =     2M13245 + M13425 + M14235 + M23145 + M31245 + M13452 + M15234 + M24135
+ M31425 + M25134 + M31452 + 2M34125 + M34152 + M35124.
sigma(M(3), M(2))    =     2M12435 + M12453 + M12534 + M13425 + M14235 + M13524 + M14253 + M23415
+ M41235 + 2M14523 + M23514 + M41253 + M24513 + M41523.
sigma(M(4), M(1))    =     2M12354 + M12453 + M12534 + M13452 + M15234.
Cocycles of total degree 6
sigma(M(1), M(5))    =     2M213456 + M231456 + M312456 + M234156 + M412356 + M234516 + M512346.
sigma(M(2), M(4))    =     2M132456 + M134256 + M142356 + M231456 + M312456 + M134526 + M152346 + M241356
+ M314256 + M134562 + M162345 + M251346 + M314526 + 2M341256 + M261345 + M314562
+ M341526 + M351246 + M341562 + M345126 + M361245 + M451236 + M345162 + M461235.
sigma(M(3), M(3))    =     2M124356 + M124536 + M125346 + M134256 + M142356 + M124563 + M126345 + M135246
+ M142536 + M234156 + M412356 + M136245 + M142563 + 2M145236 + M235146 + M412536
+ M145263 + M146235 + M236145 + M245136 + M412563 + M415236 + M145623 + M156234
+ M246135 + M345126 + M415263 + M451236 + M256134 + M346125 + M415623 + M451263
+ M356124 + M451623.
sigma(M(4), M(2))    =     2M123546 + M123564 + M123645 + M124536 + M125346 + M124635 + M125364 + M134526
+ M152346 + 2M125634 + M134625 + M152364 + M234516 + M512346 + M135624 + M152634
+ M234615 + M512364 + M145623 + M156234 + M235614 + M512634 + M245613 + M516234.
sigma(M(5), M(1))    =     2M123465 + M123564 + M123645 + M124563 + M126345 + M134562 + M162345.
Created 5 August 2002 by