Antipode on the Monomial basis of SSym

Antipodes of total degree 1
S(M1)    =     -M1.
Antipodes of total degree 2
S(M12)    =     -M12.
S(M21)    =     M12 + M21.
Antipodes of total degree 3
S(M123)    =     -M123.
S(M132)    =     -M132.
S(M213)    =     -M213.
S(M231)    =     M123 + 2M132 + M312.
S(M312)    =     M123 + 2M213 + M231.
S(M321)    =     -M123-M132 -M213-M231 -M312-M321.
Antipodes of total degree 4
S(M1234)    =     -M1234.
S(M1243)    =     -M1243.
S(M1324)    =     -M1324.
S(M1342)    =     -M1342.
S(M1423)    =     -M1423.
S(M1432)    =     -M1432.
S(M2134)    =     -M2134.
S(M2143)    =     -M2143.
S(M2314)    =     -M2314.
S(M2341)    =     M1234 + 2M1243 + M1342 + M1423 + M4123.
S(M2413)    =     -M2413.
S(M2431)    =     M1324 + M1342 + M1423 + 3M1432 + M4132.
S(M3124)    =     -M3124.
S(M3142)    =     -M3142.
S(M3214)    =     -M3214.
S(M3241)    =     M2134 + 2M2143 + M2413 + M3142 + M4213.
S(M3412)    =     M1234 + 2M1324 + M1342 + M1423 + M2314 + M3124 + M2413 + M3142
+ M3412.
S(M3421)    =     -M1234-M1243 -2M1324-2M1342 -2M1423-M3124 -3M1432-M3142
-M4123-M3412 -2M4132-M4312.
S(M4123)    =     M1234 + 2M2134 + M2314 + M3124 + M2341.
S(M4132)    =     M1243 + 2M2143 + M2413 + M3142 + M2431.
S(M4213)    =     M1324 + M2314 + M3124 + 3M3214 + M3241.
S(M4231)    =     -M1234-2M1243 -2M2134-M1423 -4M2143-M2314 -M2341-3M2413
-M3142-M4123 -2M2431-2M4213 -M4231.
S(M4312)    =     -M1234-2M1324 -M2134-M1342 -2M2314-2M3124 -M2341-M3142
-3M3214-2M3241 -M3412-M3421.
S(M4321)    =     M1234 + M1243 + M1324 + M2134 + M1342 + M1423 + M2143 + M2314
+ M3124 + M1432 + M2341 + M2413 + M3142 + M3214 + M4123 + M2431
+ M3241 + M3412 + M4132 + M4213 + M3421 + M4231 + M4312 + M4321.
Antipodes of total degree 5
S(M12345)    =     -M12345.
S(M12354)    =     -M12354.
S(M12435)    =     -M12435.
S(M12453)    =     -M12453.
S(M12534)    =     -M12534.
S(M12543)    =     -M12543.
S(M13245)    =     -M13245.
S(M13254)    =     -M13254.
S(M13425)    =     -M13425.
S(M13452)    =     -M13452.
S(M13524)    =     -M13524.
S(M13542)    =     -M13542.
S(M14235)    =     -M14235.
S(M14253)    =     -M14253.
S(M14325)    =     -M14325.
S(M14352)    =     -M14352.
S(M14523)    =     -M14523.
S(M14532)    =     -M14532.
S(M15234)    =     -M15234.
S(M15243)    =     -M15243.
S(M15324)    =     -M15324.
S(M15342)    =     -M15342.
S(M15423)    =     -M15423.
S(M15432)    =     -M15432.
S(M21345)    =     -M21345.
S(M21354)    =     -M21354.
S(M21435)    =     -M21435.
S(M21453)    =     -M21453.
S(M21534)    =     -M21534.
S(M21543)    =     -M21543.
S(M23145)    =     -M23145.
S(M23154)    =     -M23154.
S(M23415)    =     -M23415.
S(M23451)    =     M12345 + 2M12354 + M12453 + M12534 + M13452 + M15234 + M51234.
S(M23514)    =     -M23514.
S(M23541)    =     M12435 + M12453 + M12534 + 3M12543 + M13542 + M15243 + M51243.
S(M24135)    =     -M24135.
S(M24153)    =     -M24153.
S(M24315)    =     -M24315.
S(M24351)    =     M13245 + 2M13254 + M13524 + M14253 + M14352 + M15324 + M51324.
S(M24513)    =     -M24513.
S(M24531)    =     M13425 + M13452 + M13524 + 2M13542 + M14523 + 2M14532 + M15342 + M51342.
S(M25134)    =     -M25134.
S(M25143)    =     -M25143.
S(M25314)    =     -M25314.
S(M25341)    =     M14235 + M14253 + M15234 + M14523 + 2M15243 + M15342 + 2M15423 + M51423.
S(M25413)    =     -M25413.
S(M25431)    =     M14325 + M14352 + M15324 + M14532 + M15342 + M15423 + 4M15432 + M51432.
S(M31245)    =     -M31245.
S(M31254)    =     -M31254.
S(M31425)    =     -M31425.
S(M31452)    =     -M31452.
S(M31524)    =     -M31524.
S(M31542)    =     -M31542.
S(M32145)    =     -M32145.
S(M32154)    =     -M32154.
S(M32415)    =     -M32415.
S(M32451)    =     M21345 + 2M21354 + M21453 + M21534 + M25134 + M31452 + M52134.
S(M32514)    =     -M32514.
S(M32541)    =     M21435 + M21453 + M21534 + 3M21543 + M25143 + M31542 + M52143.
S(M34125)    =     -M34125.
S(M34152)    =     -M34152.
S(M34215)    =     -M34215.
S(M34251)    =     M23145 + 2M23154 + M23514 + M24153 + M25314 + M34152 + M52314.
S(M34512)    =     M12345 + 2M12435 + M12453 + M12534 + M13425 + M14235 + M13524 + M14253
+ M23415 + M41235 + 2M14523 + M23514 + M41253 + M24513 + M41523 + M45123.
S(M34521)    =     -M12345-M12354 -2M12435-2M12453 -2M12534-M13425 -M14235-3M12543
-M13452-M13524 -M14253-M15234 -M41235-2M13542 -2M14523-2M15243
-M41253-M51234 -M14532-M15342 -M15423-M41523 -2M51243-M45123
-M51342-M51423 -M54123.
S(M35124)    =     -M35124.
S(M35142)    =     -M35142.
S(M35214)    =     -M35214.
S(M35241)    =     M24135 + M24153 + M25134 + M24513 + 2M25143 + 2M25413 + M35142 + M52413.
S(M35412)    =     M13245 + M13425 + M14235 + M13452 + 3M14325 + M15234 + 2M14352 + 2M15324
+ M24315 + M41325 + M14532 + M15342 + M15423 + M25314 + M41352 + M25413
+ M41532 + M45132.
S(M35421)    =     -M13245-M13254 -M13425-M14235 -M13452-M13524 -M14253-3M14325
-M15234-M13542 -3M14352-M14523 -M15243-3M15324 -M41325-3M14532
-3M15342-3M15423 -M41352-M51324 -6M15432-M41532 -M51342-M51423
-M45132-3M51432 -M54132.
S(M41235)    =     -M41235.
S(M41253)    =     -M41253.
S(M41325)    =     -M41325.
S(M41352)    =     -M41352.
S(M41523)    =     -M41523.
S(M41532)    =     -M41532.
S(M42135)    =     -M42135.
S(M42153)    =     -M42153.
S(M42315)    =     -M42315.
S(M42351)    =     M31245 + 2M31254 + M31524 + M41253 + M35124 + M41352 + M53124.
S(M42513)    =     -M42513.
S(M42531)    =     M31425 + M31452 + M31524 + 2M31542 + M41523 + M35142 + 2M41532 + M53142.
S(M43125)    =     -M43125.
S(M43152)    =     -M43152.
S(M43215)    =     -M43215.
S(M43251)    =     M32145 + 2M32154 + M32514 + M42153 + M35214 + M43152 + M53214.
S(M43512)    =     M21345 + 2M21435 + M21453 + M21534 + M24135 + M31425 + M24153 + M31524
+ M32415 + M42135 + M24513 + M32514 + M41523 + M42153 + 2M42513 + M45213.
S(M43521)    =     -M21345-M21354 -2M21435-2M21453 -2M21534-M24135 -M31425-3M21543
-M24153-M25134 -M31452-M31524 -M42135-M24513 -2M25143-2M31542
-M41523-M42153 -M52134-M25413 -M35142-M41532 -M42513-2M52143
-M45213-M52413 -M53142-M54213.
S(M45123)    =     M12345 + 2M13245 + M13425 + M14235 + M23145 + M31245 + M13452 + M15234
+ M24135 + M31425 + M25134 + M31452 + 2M34125 + M34152 + M35124 + M34512.
S(M45132)    =     M12354 + 2M13254 + M13524 + M14253 + M23154 + M31254 + M13542 + M15243
+ M24153 + M31524 + M25143 + M31542 + M34152 + M35124 + 2M35142 + M35412.
S(M45213)    =     M12435 + M13425 + M14235 + 3M14325 + M23415 + M41235 + M14352 + M15324
+ 2M24315 + 2M41325 + M25314 + M34215 + M41352 + M42315 + M43125 + M35214
+ M43152 + M43512.
S(M45231)    =     -M12345-2M12354 -2M13245-M12534 -4M13254-M13425 -M14235-M23145
-M31245-M13452 -3M13524-2M14253 -2M15234-2M23154 -M24135-2M31254
-M31425-2M13542 -M14523-2M15243 -2M15324-M23514 -2M24153-M25134
-M31452-2M31524 -M34125-M51234 -M15342-2M15423 -M24513-2M25143
-M25314-2M31542 -2M34152-2M35124 -2M51324-2M25413 -M34512-3M35142
-M51342-M51423 -M52314-M53124 -2M35412-M52413 -M53142-M53412.
S(M45312)    =     -M12345-2M12435 -2M13245-M12453 -3M13425-3M14235 -M31245-2M13452
-M14253-6M14325 -M15234-M23415 -M31425-2M41235 -4M14352-M14523
-2M15324-2M24315 -M31452-M34125 -M41253-4M41325 -2M14532-M15342
-M25314-M34152 -3M41352-M41523 -M42315-2M43125 -M34512-2M41532
-2M43152-M45123 -2M43512-2M45132 -M45312.
S(M45321)    =     M12345 + M12354 + M12435 + 2M13245 + M12453 + M12534 + 2M13254 + 2M13425
+ 2M14235 + M31245 + M12543 + 2M13452 + 2M13524 + 2M14253 + 3M14325 + 2M15234
+ M31254 + M31425 + M41235 + 2M13542 + 3M14352 + 2M14523 + 2M15243 + 3M15324
+ M31452 + M31524 + M34125 + M41253 + 2M41325 + M51234 + 3M14532 + 3M15342
+ 3M15423 + M31542 + M34152 + M35124 + 2M41352 + M41523 + M43125 + M51243
+ 2M51324 + 4M15432 + M34512 + M35142 + 2M41532 + M43152 + M45123 + 2M51342
+ 2M51423 + M53124 + M35412 + M43512 + 2M45132 + 3M51432 + M53142 + M54123
+ M45312 + M53412 + 2M54132 + M54312.
S(M51234)    =     M12345 + 2M21345 + M23145 + M31245 + M23415 + M41235 + M23451.
S(M51243)    =     M12354 + 2M21354 + M23154 + M31254 + M23514 + M41253 + M23541.
S(M51324)    =     M12435 + 2M21435 + M24135 + M31425 + M24315 + M41325 + M24351.
S(M51342)    =     M12453 + 2M21453 + M24153 + M31452 + M24513 + M41352 + M24531.
S(M51423)    =     M12534 + 2M21534 + M25134 + M31524 + M25314 + M41523 + M25341.
S(M51432)    =     M12543 + 2M21543 + M25143 + M31542 + M25413 + M41532 + M25431.
S(M52134)    =     M13245 + M23145 + M31245 + 3M32145 + M32415 + M42135 + M32451.
S(M52143)    =     M13254 + M23154 + M31254 + 3M32154 + M32514 + M42153 + M32541.
S(M52314)    =     M13425 + M23415 + M31425 + 2M32415 + M34125 + 2M34215 + M42315 + M34251.
S(M52341)    =     -M12345-2M12354 -2M21345-M12453 -M12534-4M21354 -M23145-M31245
-M15234-2M21453 -2M21534-2M23154 -M23415-2M31254 -M23451-2M23514
-M24153-2M25134 -M31452-M31524 -M41253-M51234 -2M23541-M24513
-M25314-M35124 -M41352-2M52134 -M24531-M25341 -M52314-M53124
-M52341.
S(M52413)    =     M13524 + M23514 + M31524 + 2M32514 + M35124 + 2M35214 + M42513 + M35241.
S(M52431)    =     -M12435-M12453 -M12534-2M21435 -3M12543-2M21453 -2M21534-M24135
-M31425-M15243 -6M21543-M24153 -M24315-M25134 -M31452-M31524
-M24351-M24513 -4M25143-M25314 -3M31542-M41523 -M51243-M24531
-M25341-3M25413 -M35142-2M41532 -2M52143-3M25431 -M52413-M53142
-M52431.
S(M53124)    =     M14235 + M24135 + M41235 + M34125 + 2M42135 + M42315 + 2M43125 + M42351.
S(M53142)    =     M14253 + M24153 + M41253 + M34152 + 2M42153 + M42513 + 2M43152 + M42531.
S(M53214)    =     M14325 + M24315 + M41325 + M34215 + M42315 + M43125 + 4M43215 + M43251.
S(M53241)    =     -M13245-2M13254 -M23145-M31245 -M13524-M14253 -2M23154-2M31254
-3M32145-M15324 -M23514-M24153 -M31524-6M32154 -M32415-M41253
-M25314-M32451 -4M32514-M34152 -M35124-3M42153 -M51324-2M32541
-3M35214-M42513 -2M43152-M52314 -M53124-M35241 -M42531-3M53214
-M53241.
S(M53412)    =     -M12345-2M12435 -2M21345-M12453 -M12534-M13425 -M14235-4M21435
-M23145-M13524 -M14253-2M21453 -2M21534-2M23415 -3M24135-2M31425
-M41235-M14523 -M23451-M23514 -2M24153-2M24315 -M25134-2M31524
-2M32415-M34125 -M41253-2M42135 -2M24351-2M24513 -M25314-2M32514
-2M34215-M35124 -2M41523-2M42153 -M42315-M24531 -M25341-M34251
-2M35214-M42351 -3M42513-M45123 -M35241-M42531 -2M45213-M45231.
S(M53421)    =     M12345 + M12354 + 2M12435 + 2M21345 + 2M12453 + 2M12534 + M14235 + 2M21354
+ 4M21435 + M23145 + 3M12543 + M14253 + M15234 + 4M21453 + 4M21534 + M23154
+ M23415 + 3M24135 + M31425 + M41235 + M14523 + 2M15243 + 6M21543 + M23451
+ M23514 + 3M24153 + 2M24315 + 3M25134 + M31452 + M31524 + M41253 + 2M42135
+ M51234 + M15423 + M23541 + 2M24351 + 3M24513 + 5M25143 + 2M25314 + 2M31542
+ 2M41523 + 2M42153 + M42315 + 2M51243 + 2M52134 + 2M24531 + 2M25341 + 4M25413
+ M35142 + M41532 + M42351 + 2M42513 + M45123 + M51423 + 4M52143 + M52314
+ 3M25431 + M42531 + 2M45213 + M52341 + 3M52413 + M53142 + M54123 + M45231
+ 2M52431 + 2M54213 + M54231.
S(M54123)    =     -M12345-2M13245 -M21345-M13425 -M14235-2M23145 -2M31245-M13452
-M23415-M24135 -M31425-3M32145 -M41235-M23451 -M31452-2M32415
-2M34125-2M42135 -2M32451-M34152 -M34215-M42315 -M43125-M34251
-M34512-M42351 -M34521.
S(M54132)    =     -M12354-2M13254 -M21354-M13524 -M14253-2M23154 -2M31254-M13542
-M23514-M24153 -M31524-3M32154 -M41253-M23541 -M31542-2M32514
-M34152-M35124 -2M42153-2M32541 -M35142-M35214 -M42513-M43152
-M35241-M35412 -M42531-M35421.
S(M54213)    =     -M12435-M13425 -M14235-M21435 -3M14325-M23415 -M24135-M31425
-M41235-M14352 -3M24315-M32415 -M34125-3M41325 -M42135-M24351
-3M34215-M41352 -3M42315-3M43125 -M34251-M42351 -M43152-6M43215
-3M43251-M43512 -M43521.
S(M54231)    =     M12345 + 2M12354 + 2M13245 + M21345 + M12534 + 4M13254 + M13425 + 2M21354
+ 2M23145 + 2M31245 + M13452 + 3M13524 + M14253 + M15234 + M21534 + 4M23154
+ M23415 + 4M31254 + M31425 + 3M32145 + 2M13542 + 2M15324 + M23451 + 3M23514
+ M24153 + M25134 + M31452 + 3M31524 + 6M32154 + 2M32415 + M34125 + M41253
+ M51234 + M15342 + 2M23541 + 2M25314 + 2M31542 + 2M32451 + 5M32514 + 2M34152
+ M34215 + 3M35124 + 2M42153 + 2M51324 + M52134 + M25341 + 4M32541 + M34251
+ M34512 + 2M35142 + 4M35214 + M42513 + M43152 + M51342 + 2M52314 + 2M53124
+ M34521 + 3M35241 + 2M35412 + M42531 + M52341 + M53142 + 3M53214 + 2M35421
+ 2M53241 + M53412 + M53421.
S(M54312)    =     M12345 + 2M12435 + M13245 + M21345 + M12453 + 2M13425 + 2M14235 + 2M21435
+ M23145 + M31245 + M13452 + M14253 + 3M14325 + M21453 + 2M23415 + 2M24135
+ 2M31425 + M32145 + 2M41235 + 2M14352 + M14523 + M23451 + M24153 + 3M24315
+ M31452 + 2M32415 + 2M34125 + M41253 + 3M41325 + 2M42135 + M14532 + 2M24351
+ M24513 + M32451 + M34152 + 3M34215 + 2M41352 + M41523 + M42153 + 3M42315
+ 3M43125 + M24531 + 2M34251 + M34512 + M41532 + 2M42351 + M42513 + 2M43152
+ 4M43215 + M45123 + M34521 + M42531 + 3M43251 + 2M43512 + M45132 + M45213
+ 2M43521 + M45231 + M45312 + M45321.
S(M54321)    =     -M12345-M12354 -M12435-M12453 -M12534-M12543 -M13245-M13254
-M13425-M13452 -M13524-M13542 -M14235-M14253 -M14325-M14352
-M14523-M14532 -M15234-M15243 -M15324-M15342 -M15423-M15432
-M21345-M21354 -M21435-M21453 -M21534-M21543 -M23145-M23154
-M23415-M23451 -M23514-M23541 -M24135-M24153 -M24315-M24351
-M24513-M24531 -M25134-M25143 -M25314-M25341 -M25413-M25431
-M31245-M31254 -M31425-M31452 -M31524-M31542 -M32145-M32154
-M32415-M32451 -M32514-M32541 -M34125-M34152 -M34215-M34251
-M34512-M34521 -M35124-M35142 -M35214-M35241 -M35412-M35421
-M41235-M41253 -M41325-M41352 -M41523-M41532 -M42135-M42153
-M42315-M42351 -M42513-M42531 -M43125-M43152 -M43215-M43251
-M43512-M43521 -M45123-M45132 -M45213-M45231 -M45312-M45321
-M51234-M51243 -M51324-M51342 -M51423-M51432 -M52134-M52143
-M52314-M52341 -M52413-M52431 -M53124-M53142 -M53214-M53241
-M53412-M53421 -M54123-M54132 -M54213-M54231 -M54312-M54321.
Created 5 August 2002