Ehrhart polynomials of the root polytopes

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Polynomials

type

$L_\Phi(t)$

A2:

3*t^2 + 3*t + 1

A3:

10/3*t^3 + 5*t^2 + 11/3*t + 1

A4:

35/12*t^4 + 35/6*t^3 + 85/12*t^2 + 25/6*t + 1

A5:

21/10*t^5 + 21/4*t^4 + 28/3*t^3 + 35/4*t^2 + 137/30*t + 1

A6:

77/60*t^6 + 77/20*t^5 + 28/3*t^4 + 49/4*t^3 + 623/60*t^2 + 49/10*t + 1

A7:

143/210*t^7 + 143/60*t^6 + 451/60*t^5 + 77/6*t^4 + 937/60*t^3 + 707/60*t^2 + 363/70*t + 1

A8:

143/448*t^8 + 143/112*t^7 + 2431/480*t^6 + 429/40*t^5 + 3355/192*t^4 + 297/16*t^3 + 22079/1680*t^2 + 761/140*t + 1

A9:

2431/18144*t^9 + 2431/4032*t^8 + 4433/1512*t^7 + 715/96*t^6 + 67067/4320*t^5 + 4147/192*t^4 + 197329/9072*t^3 + 14465/1008*t^2 + 7129/1260*t + 1

A10:

46189/907200*t^10 + 46189/181440*t^9 + 89947/60480*t^8 + 26741/6048*t^7 + 493207/43200*t^6 + 171457/8640*t^5 + 4812379/181440*t^4 + 111683/4536*t^3 + 43461/2800*t^2 + 7381/1260*t + 1

A11:

4199/237600*t^11 + 4199/43200*t^10 + 121771/181440*t^9 + 46189/20160*t^8 + 540787/75600*t^7 + 216359/14400*t^6 + 14677/576*t^5 + 267839/8640*t^4 + 1567033/56700*t^3 + 417989/25200*t^2 + 83711/13860*t + 1

A12:

96577/17107200*t^12 + 96577/2851200*t^11 + 424099/1555200*t^10 + 54587/51840*t^9 + 14154829/3628800*t^8 + 5840809/604800*t^7 + 31310617/1555200*t^6 + 1588769/51840*t^5 + 3505697/97200*t^4 + 1970033/64800*t^3 + 14653847/831600*t^2 + 86021/13860*t + 1

A13:

7429/4447872*t^13 + 7429/684288*t^12 + 215441/2138400*t^11 + 676039/1555200*t^10 + 48773/25920*t^9 + 558467/103680*t^8 + 2292977/170100*t^7 + 38643397/1555200*t^6 + 1148231/31104*t^5 + 49504/1215*t^4 + 23707447/712800*t^3 + 2206841/118800*t^2 + 1145993/180180*t + 1

A14:

7429/16144128*t^14 + 7429/2306304*t^13 + 7429/217728*t^12 + 37145/228096*t^11 + 1106921/1360800*t^10 + 274873/103680*t^9 + 19862239/2540160*t^8 + 823973/48384*t^7 + 336967489/10886400*t^6 + 2215457/51840*t^5 + 137337271/2993760*t^4 + 1023961/28512*t^3 + 134187427/6879600*t^2 + 1171733/180180*t + 1

A15:

215441/1816214400*t^15 + 215441/242161920*t^14 + 1106921/103783680*t^13 + 7429/133056*t^12 + 12711019/39916800*t^11 + 141151/120960*t^10 + 6767819/1693440*t^9 + 854335/84672*t^8 + 79531321/3628800*t^7 + 8866673/241920*t^6 + 197554229/3991680*t^5 + 3370505/66528*t^4 + 17539921921/454053600*t^3 + 102778583/5045040*t^2 + 1195757/180180*t + 1

A16:

6678671/232475443200*t^16 + 6678671/29059430400*t^15 + 35978647/11623772160*t^14 + 3662497/207567360*t^13 + 20778913/182476800*t^12 + 74133991/159667200*t^11 + 49425137/27095040*t^10 + 5935771/1128960*t^9 + 21787986641/1625702400*t^8 + 769250663/29030400*t^7 + 79613363/1824768*t^6 + 445666033/7983360*t^5 + 810903015919/14529715200*t^4 + 74784723919/1816214400*t^3 + 28544683/1345344*t^2 + 2436559/360360*t + 1

A17:

392863/59880038400*t^17 + 392863/7044710400*t^16 + 12178753/14529715200*t^15 + 6678671/1291530240*t^14 + 311084131/8302694400*t^13 + 7971317/47308800*t^12 + 1680259319/2235340800*t^11 + 66184961/27095040*t^10 + 216797011/30105600*t^9 + 2983999001/180633600*t^8 + 940690133/29030400*t^7 + 10794337/215040*t^6 + 457396598173/7264857600*t^5 + 293762879477/4843238400*t^4 + 79488260179/1816214400*t^3 + 444222869/20180160*t^2 + 42142223/6126120*t + 1

A18:

392863/277159034880*t^18 + 392863/30795448320*t^17 + 40464889/190207180800*t^16 + 7464397/5283532800*t^15 + 2388214177/209227898880*t^14 + 37321985/664215552*t^13 + 884689457/3135283200*t^12 + 280035281/273715200*t^11 + 10086183307/2926264320*t^10 + 2938207519/325140480*t^9 + 303449809609/14631321600*t^8 + 275651297/7257600*t^7 + 118080490027/2037934080*t^6 + 7891279139/113218560*t^5 + 2867680552277/43589145600*t^4 + 27976829623/605404800*t^3 + 426449585/18712512*t^2 + 14274301/2042040*t + 1

A19:

765049/2633010831360*t^19 + 765049/277159034880*t^18 + 4693679/92386344960*t^17 + 392863/1086898176*t^16 + 38479897/11887948800*t^15 + 329612057/19020718080*t^14 + 249095819/2561974272*t^13 + 537829447/1379524608*t^12 + 2440508311/1642291200*t^11 + 52529585/11943936*t^10 + 34063781701/2926264320*t^9 + 14477836657/585252864*t^8 + 228515733479/5094835200*t^7 + 133032387343/2037934080*t^6 + 970367401/12579840*t^5 + 1599852193/22643712*t^4 + 167030812217/3430627200*t^3 + 24212016407/1029188160*t^2 + 275295799/38798760*t + 1

A20:

765049/13502619648000*t^20 + 765049/1350261964800*t^19 + 63499067/5543180697600*t^18 + 765049/8798699520*t^17 + 233257237/271724544000*t^16 + 6430547/1293926400*t^15 + 3924680693/126804787200*t^14 + 740257277/5434490880*t^13 + 803503815799/1379524608000*t^12 + 468561497/243302400*t^11 + 6938782991/1194393600*t^10 + 561941497/39813120*t^9 + 2185609953859/73156608000*t^8 + 26848597981/522547200*t^7 + 1501922062129/20379340800*t^6 + 2723780773/32348160*t^5 + 3645282797269/48117888000*t^4 + 487295411/9547200*t^3 + 474098765753/19554575040*t^2 + 55835135/7759752*t + 1

B2:

4*t^2 + 4*t + 1

B3:

20/3*t^3 + 8*t^2 + 10/3*t + 1

B4:

8*t^4 + 32/3*t^3 + 8*t^2 + 16/3*t + 1

B5:

36/5*t^5 + 32/3*t^4 + 44/3*t^3 + 40/3*t^2 + 62/15*t + 1

B6:

232/45*t^6 + 128/15*t^5 + 176/9*t^4 + 64/3*t^3 + 508/45*t^2 + 92/15*t + 1

B7:

968/315*t^7 + 256/45*t^6 + 884/45*t^5 + 224/9*t^4 + 1016/45*t^3 + 784/45*t^2 + 494/105*t + 1

B8:

496/315*t^8 + 1024/315*t^7 + 704/45*t^6 + 1024/45*t^5 + 1472/45*t^4 + 1408/45*t^3 + 4432/315*t^2 + 704/105*t + 1

B9:

2012/2835*t^9 + 512/315*t^8 + 9736/945*t^7 + 256/15*t^6 + 964/27*t^5 + 608/15*t^4 + 85432/2835*t^3 + 6544/315*t^2 + 1622/315*t + 1

B10:

1352/4725*t^10 + 2048/2835*t^9 + 608/105*t^8 + 2048/189*t^7 + 2312/75*t^6 + 5504/135*t^5 + 44048/945*t^4 + 22976/567*t^3 + 25964/1575*t^2 + 2252/315*t + 1

B11:

776/7425*t^11 + 4096/14175*t^10 + 1612/567*t^9 + 5632/945*t^8 + 103616/4725*t^7 + 22528/675*t^6 + 2444/45*t^5 + 160864/2835*t^4 + 528716/14175*t^3 + 4136/175*t^2 + 19102/3465*t + 1

B12:

16336/467775*t^12 + 16384/155925*t^11 + 52768/42525*t^10 + 8192/2835*t^9 + 26848/2025*t^8 + 108544/4725*t^7 + 2126464/42525*t^6 + 34816/567*t^5 + 2587016/42525*t^4 + 696224/14175*t^3 + 138296/7425*t^2 + 26032/3465*t + 1

B13:

65432/6081075*t^13 + 16384/467775*t^12 + 228232/467775*t^11 + 53248/42525*t^10 + 32948/4725*t^9 + 193024/14175*t^8 + 1609856/42525*t^7 + 2289664/42525*t^6 + 3179224/42525*t^5 + 3109184/42525*t^4 + 6874684/155925*t^3 + 1358552/51975*t^2 + 262186/45045*t + 1

B14:

26192/8513505*t^14 + 65536/6081075*t^13 + 27232/155925*t^12 + 32768/66825*t^11 + 138088/42525*t^10 + 14336/2025*t^9 + 7232416/297675*t^8 + 1681408/42525*t^7 + 342896/4725*t^6 + 509696/6075*t^5 + 35155936/467775*t^4 + 1273216/22275*t^3 + 97207532/4729725*t^2 + 352276/45045*t + 1

B15:

74864/91216125*t^15 + 131072/42567525*t^14 + 1047016/18243225*t^13 + 16384/93555*t^12 + 9523376/7016625*t^11 + 139264/42525*t^10 + 1725916/127575*t^9 + 1483264/59535*t^8 + 36906224/637875*t^7 + 3309056/42525*t^6 + 135521912/1403325*t^5 + 8368064/93555*t^4 + 512215024/10135125*t^3 + 134196512/4729725*t^2 + 274198/45045*t + 1

B16:

32/155925*t^16 + 524288/638512875*t^15 + 57088/3274425*t^14 + 1048576/18243225*t^13 + 2432/4725*t^12 + 9568256/7016625*t^11 + 56576/8505*t^10 + 12255232/893025*t^9 + 11622592/297675*t^8 + 38457344/637875*t^7 + 15283456/155925*t^6 + 151212032/1403325*t^5 + 8368064/93555*t^4 + 4591979008/70945875*t^3 + 737824/33075*t^2 + 364288/45045*t + 1

B17:

34948/723647925*t^17 + 131072/638512875*t^16 + 3144368/638512875*t^15 + 2228224/127702575*t^14 + 3259096/18243225*t^13 + 3620864/7016625*t^12 + 143830832/49116375*t^11 + 5988352/893025*t^10 + 4075676/178605*t^9 + 178545152/4465125*t^8 + 81857872/1002375*t^7 + 147001856/1403325*t^6 + 15248029976/127702575*t^5 + 22481329856/212837625*t^4 + 4021170928/70945875*t^3 + 143759072/4729725*t^2 + 4841546/765765*t + 1

B18:

1048504/97692469875*t^18 + 524288/10854718875*t^17 + 2489792/1915538625*t^16 + 1048576/212837625*t^15 + 109492528/1915538625*t^14 + 5439488/30405375*t^13 + 515792864/442047375*t^12 + 144474112/49116375*t^11 + 157472744/13395375*t^10 + 34428928/1488375*t^9 + 8477648672/147349125*t^8 + 198658048/2338875*t^7 + 723686657648/5746615875*t^6 + 84670648576/638512875*t^5 + 66169148576/638512875*t^4 + 5099363968/70945875*t^3 + 5770309052/241215975*t^2 + 6373076/765765*t + 1

B19:

4194152/1856156927625*t^19 + 1048576/97692469875*t^18 + 10484468/32564156625*t^17 + 2490368/1915538625*t^16 + 6498496/383107725*t^15 + 9961472/174139875*t^14 + 2439899944/5746615875*t^13 + 47005696/40186125*t^12 + 797489864/147349125*t^11 + 158683136/13395375*t^10 + 5166641884/147349125*t^9 + 8673202688/147349125*t^8 + 626381137792/5746615875*t^7 + 771313897472/5746615875*t^6 + 91311144152/638512875*t^5 + 7058428864/58046625*t^4 + 75431804836/1206079875*t^3 + 707075272/21928725*t^2 + 95052434/14549535*t + 1

B20:

4194224/9280784638125*t^20 + 4194304/1856156927625*t^19 + 667232/8881133625*t^18 + 2097152/6512831325*t^17 + 14982592/3192564375*t^16 + 32505856/1915538625*t^15 + 116681216/820945125*t^14 + 488636416/1149323175*t^13 + 4976278576/2210236875*t^12 + 800899072/147349125*t^11 + 102664736/5457375*t^10 + 1046880256/29469825*t^9 + 208267282208/2612098125*t^8 + 650484697088/5746615875*t^7 + 896321796224/5746615875*t^6 + 2244810752/14189175*t^5 + 6386159144504/54273594375*t^4 + 3513939808/44669625*t^3 + 23298510568/916620705*t^2 + 124151504/14549535*t + 1

C3:

16/3*t^3 + 8*t^2 + 14/3*t + 1

C4:

16/3*t^4 + 32/3*t^3 + 32/3*t^2 + 16/3*t + 1

C5:

64/15*t^5 + 32/3*t^4 + 16*t^3 + 40/3*t^2 + 86/15*t + 1

C6:

128/45*t^6 + 128/15*t^5 + 160/9*t^4 + 64/3*t^3 + 692/45*t^2 + 92/15*t + 1

C7:

512/315*t^7 + 256/45*t^6 + 704/45*t^5 + 224/9*t^4 + 1184/45*t^3 + 784/45*t^2 + 674/105*t + 1

C8:

256/315*t^8 + 1024/315*t^7 + 512/45*t^6 + 1024/45*t^5 + 1472/45*t^4 + 1408/45*t^3 + 6016/315*t^2 + 704/105*t + 1

C9:

1024/2835*t^9 + 512/315*t^8 + 6656/945*t^7 + 256/15*t^6 + 4288/135*t^5 + 608/15*t^4 + 101792/2835*t^3 + 6544/315*t^2 + 2182/315*t + 1

C10:

2048/14175*t^10 + 2048/2835*t^9 + 512/135*t^8 + 2048/189*t^7 + 17024/675*t^6 + 5504/135*t^5 + 137888/2835*t^4 + 22976/567*t^3 + 4996/225*t^2 + 2252/315*t + 1

C11:

8192/155925*t^11 + 4096/14175*t^10 + 1024/567*t^9 + 5632/945*t^8 + 26624/1575*t^7 + 22528/675*t^6 + 144832/2835*t^5 + 160864/2835*t^4 + 635312/14175*t^3 + 4136/175*t^2 + 25402/3465*t + 1

C12:

8192/467775*t^12 + 16384/155925*t^11 + 32768/42525*t^10 + 8192/2835*t^9 + 138752/14175*t^8 + 108544/4725*t^7 + 1854464/42525*t^6 + 34816/567*t^5 + 2761072/42525*t^4 + 696224/14175*t^3 + 1293472/51975*t^2 + 26032/3465*t + 1

C13:

32768/6081075*t^13 + 16384/467775*t^12 + 139264/467775*t^11 + 53248/42525*t^10 + 23552/4725*t^9 + 193024/14175*t^8 + 1329152/42525*t^7 + 2289664/42525*t^6 + 3089536/42525*t^5 + 3109184/42525*t^4 + 8285488/155925*t^3 + 1358552/51975*t^2 + 345346/45045*t + 1

C14:

65536/42567525*t^14 + 65536/6081075*t^13 + 16384/155925*t^12 + 32768/66825*t^11 + 96256/42525*t^10 + 14336/2025*t^9 + 5734912/297675*t^8 + 1681408/42525*t^7 + 2304/35*t^6 + 509696/6075*t^5 + 38020672/467775*t^4 + 1273216/22275*t^3 + 128912372/4729725*t^2 + 352276/45045*t + 1

C15:

262144/638512875*t^15 + 131072/42567525*t^14 + 622592/18243225*t^13 + 16384/93555*t^12 + 6504448/7016625*t^11 + 139264/42525*t^10 + 9288704/893025*t^9 + 1483264/59535*t^8 + 31839232/637875*t^7 + 3309056/42525*t^6 + 134475904/1403325*t^5 + 8368064/93555*t^4 + 1441195328/23648625*t^3 + 134196512/4729725*t^2 + 358282/45045*t + 1

C16:

65536/638512875*t^16 + 524288/638512875*t^15 + 262144/25540515*t^14 + 1048576/18243225*t^13 + 2424832/7016625*t^12 + 9568256/7016625*t^11 + 4456448/893025*t^10 + 12255232/893025*t^9 + 144894976/4465125*t^8 + 38457344/637875*t^7 + 128100352/1403325*t^6 + 151212032/1403325*t^5 + 20759337728/212837625*t^4 + 4591979008/70945875*t^3 + 138977792/4729725*t^2 + 364288/45045*t + 1

C17:

262144/10854718875*t^17 + 131072/638512875*t^16 + 262144/91216125*t^15 + 2228224/127702575*t^14 + 1540096/13030875*t^13 + 3620864/7016625*t^12 + 3915776/1819125*t^11 + 5988352/893025*t^10 + 11754496/637875*t^9 + 178545152/4465125*t^8 + 509502464/7016625*t^7 + 147001856/1403325*t^6 + 76736061568/638512875*t^5 + 22481329856/212837625*t^4 + 692238784/10135125*t^3 + 143759072/4729725*t^2 + 6282986/765765*t + 1

C18:

524288/97692469875*t^18 + 524288/10854718875*t^17 + 131072/174139875*t^16 + 1048576/212837625*t^15 + 71499776/1915538625*t^14 + 5439488/30405375*t^13 + 372588544/442047375*t^12 + 144474112/49116375*t^11 + 124254208/13395375*t^10 + 34428928/1488375*t^9 + 7282596352/147349125*t^8 + 198658048/2338875*t^7 + 62422113536/522419625*t^6 + 84670648576/638512875*t^5 + 72543353536/638512875*t^4 + 5099363968/70945875*t^3 + 7554770332/241215975*t^2 + 6373076/765765*t + 1

C19:

2097152/1856156927625*t^19 + 1048576/97692469875*t^18 + 6029312/32564156625*t^17 + 2490368/1915538625*t^16 + 4194304/383107725*t^15 + 9961472/174139875*t^14 + 1735622656/5746615875*t^13 + 47005696/40186125*t^12 + 617160704/147349125*t^11 + 158683136/13395375*t^10 + 4321432576/147349125*t^9 + 8673202688/147349125*t^8 + 569293760512/5746615875*t^7 + 771313897472/5746615875*t^6 + 92843566208/638512875*t^5 + 7058428864/58046625*t^4 + 90782781136/1206079875*t^3 + 707075272/21928725*t^2 + 122619974/14549535*t + 1

C20:

2097152/9280784638125*t^20 + 4194304/1856156927625*t^19 + 4194304/97692469875*t^18 + 2097152/6512831325*t^17 + 9568256/3192564375*t^16 + 32505856/1915538625*t^15 + 573046784/5746615875*t^14 + 488636416/1149323175*t^13 + 3786661888/2210236875*t^12 + 800899072/147349125*t^11 + 756318208/49116375*t^10 + 1046880256/29469825*t^9 + 2016516872704/28733079375*t^8 + 650484697088/5746615875*t^7 + 862228732928/5746615875*t^6 + 2244810752/14189175*t^5 + 7026574674512/54273594375*t^4 + 3513939808/44669625*t^3 + 151689504224/4583103525*t^2 + 124151504/14549535*t + 1

D4:

4*t^4 + 8*t^3 + 8*t^2 + 4*t + 1

D5:

18/5*t^5 + 9*t^4 + 38/3*t^3 + 10*t^2 + 71/15*t + 1

D6:

116/45*t^6 + 116/15*t^5 + 136/9*t^4 + 52/3*t^3 + 554/45*t^2 + 74/15*t + 1

D7:

484/315*t^7 + 242/45*t^6 + 634/45*t^5 + 196/9*t^4 + 988/45*t^3 + 623/45*t^2 + 569/105*t + 1

D8:

248/315*t^8 + 992/315*t^7 + 32/3*t^6 + 944/45*t^5 + 144/5*t^4 + 1184/45*t^3 + 992/63*t^2 + 584/105*t + 1

D9:

1006/2835*t^9 + 503/315*t^8 + 6404/945*t^7 + 244/15*t^6 + 3946/135*t^5 + 542/15*t^4 + 87068/2835*t^3 + 5356/315*t^2 + 1867/315*t + 1

D10:

676/4725*t^10 + 676/945*t^9 + 1168/315*t^8 + 664/63*t^7 + 1796/75*t^6 + 1708/45*t^5 + 41176/945*t^4 + 6568/189*t^3 + 29342/1575*t^2 + 634/105*t + 1

D11:

388/7425*t^11 + 194/675*t^10 + 722/405*t^9 + 88/15*t^8 + 77408/4725*t^7 + 2398/75*t^6 + 6418/135*t^5 + 6908/135*t^4 + 78794/2025*t^3 + 4433/225*t^2 + 21937/3465*t + 1

D12:

8168/467775*t^12 + 16336/155925*t^11 + 32528/42525*t^10 + 232/81*t^9 + 136208/14175*t^8 + 105856/4725*t^7 + 1772864/42525*t^6 + 163112/2835*t^5 + 2499988/42525*t^4 + 86408/2025*t^3 + 1098232/51975*t^2 + 22252/3465*t + 1

D13:

32716/6081075*t^13 + 16358/467775*t^12 + 138692/467775*t^11 + 52988/42525*t^10 + 9986/2025*t^9 + 190268/14175*t^8 + 184768/6075*t^7 + 2201264/42525*t^6 + 2895212/42525*t^5 + 2826343/42525*t^4 + 1038082/22275*t^3 + 1147042/51975*t^2 + 300301/45045*t + 1

D14:

13096/8513505*t^14 + 13096/1216215*t^13 + 9808/93555*t^12 + 6536/13365*t^11 + 95668/42525*t^10 + 316/45*t^9 + 1128592/59535*t^8 + 328576/8505*t^7 + 2687864/42525*t^6 + 19192/243*t^5 + 6935696/93555*t^4 + 223292/4455*t^3 + 110417882/4729725*t^2 + 303766/45045*t + 1

D15:

37432/91216125*t^15 + 18716/6081075*t^14 + 621812/18243225*t^13 + 3272/18711*t^12 + 6482008/7016625*t^11 + 138634/42525*t^10 + 1314542/127575*t^9 + 41816/1701*t^8 + 31063672/637875*t^7 + 3189596/42525*t^6 + 126630844/1403325*t^5 + 1530376/18711*t^4 + 181921384/3378375*t^3 + 16340141/675675*t^2 + 313237/45045*t + 1

D16:

16/155925*t^16 + 128/155925*t^15 + 48512/4729725*t^14 + 116416/2027025*t^13 + 161344/467775*t^12 + 212096/155925*t^11 + 211072/42525*t^10 + 64352/4725*t^9 + 3173152/99225*t^8 + 836224/14175*t^7 + 41124992/467775*t^6 + 15871552/155925*t^5 + 8368064/93555*t^4 + 1273216/22275*t^3 + 119852672/4729725*t^2 + 63248/9009*t + 1

D17:

17474/723647925*t^17 + 8737/42567525*t^16 + 1834328/638512875*t^15 + 742424/42567525*t^14 + 2153836/18243225*t^13 + 48212/93555*t^12 + 105404312/49116375*t^11 + 397528/59535*t^10 + 16316806/893025*t^9 + 11754038/297675*t^8 + 498017944/7016625*t^7 + 47327048/467775*t^6 + 14504162444/127702575*t^5 + 4130342644/42567525*t^4 + 4306574968/70945875*t^3 + 123438632/4729725*t^2 + 5517221/765765*t + 1

D18:

524252/97692469875*t^18 + 524252/10854718875*t^17 + 1441504/1915538625*t^16 + 1048336/212837625*t^15 + 71457944/1915538625*t^14 + 5435432/30405375*t^13 + 33813968/40186125*t^12 + 144133552/49116375*t^11 + 123649012/13395375*t^10 + 34182772/1488375*t^9 + 7184819344/147349125*t^8 + 194604688/2338875*t^7 + 662829628984/5746615875*t^6 + 80207443384/638512875*t^5 + 66806569072/638512875*t^4 + 4528555888/70945875*t^3 + 595546442/21928725*t^2 + 5562266/765765*t + 1

D19:

2097076/1856156927625*t^19 + 1048538/97692469875*t^18 + 123034/664574625*t^17 + 2490064/1915538625*t^16 + 2994976/273648375*t^15 + 109532036/1915538625*t^14 + 1733981588/5746615875*t^13 + 516392488/442047375*t^12 + 3516892/841995*t^11 + 158044318/13395375*t^10 + 612507554/21049875*t^9 + 8569993624/147349125*t^8 + 557241980864/5746615875*t^7 + 746177298676/5746615875*t^6 + 4190042684/30405375*t^5 + 71587222792/638512875*t^4 + 142963838/2127125*t^3 + 6718304107/241215975*t^2 + 108070439/14549535*t + 1

D20:

2097112/9280784638125*t^20 + 4194224/1856156927625*t^19 + 599152/13956067125*t^18 + 2097016/6512831325*t^17 + 28699808/9577693125*t^16 + 4642688/273648375*t^15 + 572748544/5746615875*t^14 + 488290928/1149323175*t^13 + 540113704/315748125*t^12 + 799104752/147349125*t^11 + 2258731184/147349125*t^10 + 148535384/4209975*t^9 + 284443902032/4104725625*t^8 + 637798613248/5746615875*t^7 + 17016690752/117277875*t^6 + 19181682064/127702575*t^5 + 6492895066172/54273594375*t^4 + 12091770088/172297125*t^3 + 132135642344/4583103525*t^2 + 108836204/14549535*t + 1

E6:

39/10*t^6 + 117/10*t^5 + 75/4*t^4 + 18*t^3 + 267/20*t^2 + 63/10*t + 1

E7:

148/35*t^7 + 72/5*t^6 + 24*t^5 + 28*t^4 + 488/15*t^3 + 98/5*t^2 + 68/21*t + 1

E8:

57/7*t^8 + 108/7*t^7 + 30*t^6 + 72*t^5 + 39*t^4 + 36*t^3 + 300/7*t^2 - 24/7*t + 1

F4:

16*t^4 + 16*t^3 + 8*t^2 + 8*t + 1

G2:

9*t^2 + 3*t + 1

Definition

For an irreducible crystallographic root system $\Phi$ [5], let $P_\Phi=\operatorname{conv}(\Phi)$ be the full root polytope in the root lattice $\mathbb{Z}\Phi$. This table stores the Ehrhart polynomial $L_\Phi(t)=|tP_\Phi\cap\mathbb{Z}\Phi|$.

Parameters

type

— Dynkin type ($A_n$ with $n\geq1$, $B_n$ with $n\geq2$, $C_n$ with $n\geq3$, $D_n$ with $n\geq4$, or one of $E_6$, $E_7$, $E_8$, $F_4$, $G_2$)

Formulas

(1)

$\sum_{t\geq0}L_\Phi(t)z^t=h^*_{P_\Phi}(z)/(1-z)^{r+1}$, where $r=\operatorname{rank}(\Phi)$ [6].

(2)

If $h^*_{P_\Phi}(z)=\sum_i h_i^*z^i$, then $L_\Phi(t)=\sum_i h_i^*\binom{t+r-i}{r}$.

(3)

$L_\Phi(1)=|\Phi|+1$, since the roots and the origin are the lattice points of $P_\Phi$.

Comments

(4)

Some authors use root polytope for $\operatorname{conv}(\Phi^+\cup\{0\})$; this table uses $P_\Phi=\operatorname{conv}(\Phi)$. Counting in $\mathbb{Z}\Phi$ matters: for $C_n$ and $D_n$ in the usual coordinates, $\mathbb{Z}^n$ also contains $\pm e_i\in P_\Phi$, and counting in $\mathbb{Z}^n$ gives a different polynomial.

(5)

$n$ follows Bourbaki's ranges, which list each irreducible root system once: $C_2\cong B_2$ and $D_3\cong A_3$. $A_1$ is omitted as a row because its root polytope is a segment, with $L_{A_1}(t)=2t+1$ and $h^*_{A_1}(z)=1+z$.

Programs

(P1)

Sage

from sage.arith.misc import binomial, factorial
from sage.rings.rational_field import QQ
from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing

T = PolynomialRing(QQ, "t")
t = T.gen()

def ehrhart_from_h_star(coefficients):
    r = len(coefficients) - 1
    def choose(poly, m):
        out = T(1)
        for j in range(m):
            out *= poly - j
        return out / factorial(m)
    return sum(QQ(c)*choose(t + r - i, r) for i, c in enumerate(coefficients))

ehrhart_from_h_star([1, 16, 36, 16, 1])  # L_{A_4}(t)

References

[1]

Federico Ardila, Matthias Beck, Serkan Hosten, Julian Pfeifle and Kim Seashore, Root polytopes and growth series of root lattices, SIAM Journal on Discrete Mathematics 25 (2011), 360-378. (arXiv) (doi)

[2]

Roland Bacher, Pierre de la Harpe and Boris Venkov, Series de croissance et polynomes d'Ehrhart associes aux reseaux de racines, Annales de l'Institut Fourier 49 (1999), 727-762. (doi)

[3]

Michael Baake and Uwe Grimm, Coordination sequences for root lattices and related graphs, Zeitschrift fuer Kristallographie 212 (1997), 253-256. (arXiv)

[4]

John H. Conway and Neil J. A. Sloane, Low-dimensional lattices. VII. Coordination sequences, Proceedings of the Royal Society of London A 453 (1997), 2369-2389. (doi)

Links

[5]

Wikipedia: Root system

[6]

Wikipedia: Ehrhart polynomial

[7]

OEIS A008459: coordinator triangle for the root lattice $A_n$

[8]

OEIS A109001: coordinator triangle for the root lattice $B_n$

[9]

OEIS A108558: coordinator triangle for the root lattice $D_n$

[10]

OEIS A019557: coordination sequence for the root lattice $G_2$

[11]

OEIS A019558: coordination sequence for the root lattice $F_4$

[12]

OEIS A008399: coordination sequence for the root lattice $E_6$

[13]

OEIS A008397: coordination sequence for the root lattice $E_7$

[14]

OEIS A008340: coordination sequence for the root lattice $E_8$

Similar tables

Ehrhart $h^*$-polynomials of the root polytopes — gives the numerator $h^*_{P_\Phi}(z)$ of the Ehrhart series; the two tables determine each other by (1) and (2)

Packing densities and Hermite numbers of the classical lattices — the lattices $A_n$, $D_n$ and $E_n$, with packing and Hermite data

Covering radii and covering densities of the classical lattices — the same classical root lattices, with covering data

Kissing numbers $\tau_n$ — for simply-laced $\Phi$, $L_\Phi(1)-1=|\Phi|$ is the kissing number of $\mathbb{Z}\Phi$; for $A_2$, $A_3$, $D_4$ and $E_8$ it is $\tau_n$

Hermite's constants $\gamma_n$ — $\gamma_n$ for $2\leq n\leq8$ is attained by the root lattices $A_2$, $A_3$, $D_4$, $D_5$, $E_6$, $E_7$ and $E_8$

Data properties

Entries are of type: rational polynomial

How well the digits are known: exact

Table is complete: no (it holds $L_\Phi(t)$ for $A_n$ and $B_n$ with $2\leq n\leq20$, $C_n$ with $3\leq n\leq20$, $D_n$ with $4\leq n\leq20$, and the exceptional types $E_6$, $E_7$, $E_8$, $F_4$ and $G_2$; the range stops at $n=20$ because the longest row, $L_{A_{20}}(t)$, is 569 characters, and the $A_1$ case is given in the comments)

How they were obtained:

All entries are exact polynomials. The $h^*$-polynomials of [1], [2], [3] and [4] are converted to Ehrhart polynomials by the binomial transform in (2).

Every stored row was checked against $L_\Phi(1)=|\Phi|+1$. Direct lattice-point counts from Sage root-lattice coordinates and PPL facet inequalities matched $A_2$, $A_3$, $A_4$, $B_2$, $B_3$, $C_3$, $D_4$, $G_2$ and $F_4$ over the counted dilates.