Ehrhart polynomials of the hypersimplices

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Polynomials

$n$

$k$

$L_{\Delta(k,n)}(t)$

2/3*t^3 + 2*t^2 + 7/3*t + 1

11/24*t^4 + 25/12*t^3 + 85/24*t^2 + 35/12*t + 1

13/60*t^5 + 3/2*t^4 + 47/12*t^3 + 5*t^2 + 101/30*t + 1

11/20*t^5 + 11/4*t^4 + 23/4*t^3 + 25/4*t^2 + 37/10*t + 1

19/240*t^6 + 63/80*t^5 + 49/16*t^4 + 287/48*t^3 + 763/120*t^2 + 56/15*t + 1

151/360*t^6 + 161/60*t^5 + 259/36*t^4 + 21/2*t^3 + 3199/360*t^2 + 259/60*t + 1

1/42*t^7 + 29/90*t^6 + 53/30*t^5 + 91/18*t^4 + 49/6*t^3 + 343/45*t^2 + 283/70*t + 1

397/1680*t^7 + 359/180*t^6 + 281/40*t^5 + 245/18*t^4 + 1273/80*t^3 + 2051/180*t^2 + 2027/420*t + 1

151/315*t^7 + 151/45*t^6 + 463/45*t^5 + 161/9*t^4 + 862/45*t^3 + 574/45*t^2 + 533/105*t + 1

247/40320*t^8 + 121/1120*t^7 + 763/960*t^6 + 253/80*t^5 + 14203/1920*t^4 + 1667/160*t^3 + 88721/10080*t^2 + 1207/280*t + 1

477/4480*t^8 + 1311/1120*t^7 + 1731/320*t^6 + 1107/80*t^5 + 13899/640*t^4 + 3477/160*t^3 + 15419/1120*t^2 + 1473/280*t + 1

15619/40320*t^8 + 3607/1120*t^7 + 11311/960*t^6 + 1991/80*t^5 + 63991/1920*t^4 + 4669/160*t^3 + 166337/10080*t^2 + 1599/280*t + 1

251/181440*t^9 + 31/1008*t^8 + 1765/6048*t^7 + 37/24*t^6 + 42863/8640*t^5 + 481/48*t^4 + 115205/9072*t^3 + 4993/504*t^2 + 5729/1260*t + 1

913/22680*t^9 + 1135/2016*t^8 + 5071/1512*t^7 + 179/16*t^6 + 3128/135*t^5 + 2999/96*t^4 + 63041/2268*t^3 + 8069/504*t^2 + 3553/630*t + 1

44117/181440*t^9 + 2489/1008*t^8 + 66547/6048*t^7 + 683/24*t^6 + 409361/8640*t^5 + 2543/48*t^4 + 363947/9072*t^3 + 10127/504*t^2 + 7883/1260*t + 1

15619/36288*t^9 + 15619/4032*t^8 + 94939/6048*t^7 + 3607/96*t^6 + 101311/1728*t^5 + 11911/192*t^4 + 25394/567*t^3 + 21689/1008*t^2 + 1627/252*t + 1

1013/3628800*t^10 + 5533/725760*t^9 + 2189/24192*t^8 + 14795/24192*t^7 + 447689/172800*t^6 + 246697/34560*t^5 + 14597/1134*t^4 + 543763/36288*t^3 + 91949/8400*t^2 + 1199/252*t + 1

299/22680*t^10 + 16621/72576*t^9 + 41591/24192*t^8 + 88693/12096*t^7 + 170137/8640*t^6 + 604109/17280*t^5 + 3043997/72576*t^4 + 308473/9072*t^3 + 60929/3360*t^2 + 15059/2520*t + 1

56899/453600*t^10 + 565631/362880*t^9 + 205733/24192*t^8 + 326491/12096*t^7 + 2400629/43200*t^6 + 1348787/17280*t^5 + 5535695/72576*t^4 + 468655/9072*t^3 + 1185701/50400*t^2 + 16973/2520*t + 1

655177/1814400*t^10 + 336083/90720*t^9 + 2078791/120960*t^8 + 287639/6048*t^7 + 7525771/86400*t^6 + 95557/864*t^5 + 35914087/362880*t^4 + 1125575/18144*t^3 + 443179/16800*t^2 + 17897/2520*t + 1

509/9979200*t^11 + 169/100800*t^10 + 551/22680*t^9 + 2057/10080*t^8 + 332249/302400*t^7 + 18997/4800*t^6 + 876959/90720*t^5 + 80179/5040*t^4 + 244681/14175*t^3 + 150293/12600*t^2 + 68591/13860*t + 1

50879/13305600*t^11 + 6979/86400*t^10 + 60271/80640*t^9 + 32153/8064*t^8 + 5483809/403200*t^7 + 897259/28800*t^6 + 11875111/241920*t^5 + 185339/3456*t^4 + 451173/11200*t^3 + 338503/16800*t^2 + 58007/9240*t + 1

1093/19800*t^11 + 62879/75600*t^10 + 20893/3780*t^9 + 10813/504*t^8 + 684323/12600*t^7 + 340967/3600*t^6 + 5258/45*t^5 + 38819/378*t^4 + 1202029/18900*t^3 + 42218/1575*t^2 + 1103/154*t + 1

1623019/6652800*t^11 + 882773/302400*t^10 + 1908073/120960*t^9 + 1028401/20160*t^8 + 7395023/67200*t^7 + 2401619/14400*t^6 + 4398559/24192*t^5 + 8661917/60480*t^4 + 12163441/151200*t^3 + 782969/25200*t^2 + 8861/1155*t + 1

655177/1663200*t^11 + 655177/151200*t^10 + 5507/252*t^9 + 336083/5040*t^8 + 6898277/50400*t^7 + 1430341/7200*t^6 + 3152491/15120*t^5 + 1200463/7560*t^4 + 30291/350*t^3 + 68321/2100*t^2 + 18107/2310*t + 1

1361/159667200*t^12 + 1261/3801600*t^11 + 83551/14515200*t^10 + 85241/1451520*t^9 + 30017/76800*t^8 + 4304651/2419200*t^7 + 81998813/14515200*t^6 + 865267/69120*t^5 + 69295109/3628800*t^4 + 35417369/1814400*t^3 + 1018381/79200*t^2 + 35503/6930*t + 1

478271/479001600*t^12 + 2010749/79833600*t^11 + 12264473/43545600*t^10 + 32981/17920*t^9 + 113291321/14515200*t^8 + 54634879/2419200*t^7 + 1990504139/43545600*t^6 + 94940131/1451520*t^5 + 719975269/10886400*t^4 + 28178267/604800*t^3 + 18350293/831600*t^2 + 60541/9240*t + 1

226393/10644480*t^12 + 408395/1064448*t^11 + 8921861/2903040*t^10 + 4200209/290304*t^9 + 8649953/193536*t^8 + 46363967/483840*t^7 + 5267743/35840*t^6 + 79099241/483840*t^5 + 95648839/725760*t^4 + 27503801/362880*t^3 + 1106599/36960*t^2 + 104611/13860*t + 1

11053079/79833600*t^12 + 25873471/13305600*t^11 + 89444797/7257600*t^10 + 33963137/725760*t^9 + 31995301/268800*t^8 + 259371983/1209600*t^7 + 2043169271/7257600*t^6 + 13183313/48384*t^5 + 87953099/453600*t^4 + 90426167/907200*t^3 + 19728683/554400*t^2 + 226811/27720*t + 1

27085381/79833600*t^12 + 55168451/13305600*t^11 + 11280737/483840*t^10 + 11588317/145152*t^9 + 448950671/2419200*t^8 + 373608443/1209600*t^7 + 548740361/1451520*t^6 + 27715103/80640*t^5 + 7768501/33600*t^4 + 102484187/907200*t^3 + 4270747/110880*t^2 + 235391/27720*t + 1

1363/1037836800*t^13 + 1021/17107200*t^12 + 557/456192*t^11 + 23101/1555200*t^10 + 123949/1036800*t^9 + 347191/518400*t^8 + 3882239/1451520*t^7 + 11921923/1555200*t^6 + 4060727/259200*t^5 + 8711729/388800*t^4 + 1550689/71280*t^3 + 408161/29700*t^2 + 951953/180180*t + 1

82207/345945600*t^13 + 241177/34214400*t^12 + 1068137/11404800*t^11 + 2288767/3110400*t^10 + 1314211/345600*t^9 + 14111227/1036800*t^8 + 83790269/2419200*t^7 + 197267941/3110400*t^6 + 43426663/518400*t^5 + 15421757/194400*t^4 + 25135273/475200*t^3 + 5677373/237600*t^2 + 272401/40040*t + 1

910669/124540416*t^13 + 2666489/17107200*t^12 + 50860811/34214400*t^11 + 13054769/1555200*t^10 + 16041/512*t^9 + 42388619/518400*t^8 + 3353880181/21772800*t^7 + 331357247/1555200*t^6 + 33840991/155520*t^5 + 63442561/388800*t^4 + 15706253/178200*t^3 + 1956773/59400*t^2 + 355804/45045*t + 1

28218769/415134720*t^13 + 25502053/22809600*t^12 + 189526301/22809600*t^11 + 76834823/2073600*t^10 + 46036561/414720*t^9 + 54552121/230400*t^8 + 5388926531/14515200*t^7 + 901850989/2073600*t^6 + 79259561/207360*t^5 + 129706447/518400*t^4 + 340659967/2851200*t^3 + 395317/9900*t^2 + 3115129/360360*t + 1

125468459/518918400*t^13 + 635641/190080*t^12 + 121304801/5702400*t^11 + 4281511/51840*t^10 + 113398337/518400*t^9 + 7224113/17280*t^8 + 2156231471/3628800*t^7 + 11012027/17280*t^6 + 4188971/8100*t^5 + 4081337/12960*t^4 + 50182301/356400*t^3 + 175487/3960*t^2 + 1635857/180180*t + 1

27085381/74131200*t^13 + 27085381/5702400*t^12 + 242237/8448*t^11 + 55168451/518400*t^10 + 140480471/518400*t^9 + 9604439/19200*t^8 + 71470307/103680*t^7 + 372761233/518400*t^6 + 5463163/9600*t^5 + 5487443/16200*t^4 + 21146681/142560*t^3 + 604643/13200*t^2 + 237371/25740*t + 1

16369/87178291200*t^14 + 8179/830269440*t^13 + 6419/27371520*t^12 + 30577/9123840*t^11 + 2790833/87091200*t^10 + 179381/829440*t^9 + 129073883/121927680*t^8 + 22068727/5806080*t^7 + 62234543/6220800*t^6 + 3957619/207360*t^5 + 618362471/23950080*t^4 + 1708009/71280*t^3 + 1103807123/75675600*t^2 + 978683/180180*t + 1

252073/4843238400*t^14 + 1409/786240*t^13 + 7067/253440*t^12 + 197077/760320*t^11 + 969827/604800*t^10 + 160433/23040*t^9 + 73916287/3386880*t^8 + 24212687/483840*t^7 + 19422619/230400*t^6 + 7190401/69120*t^5 + 247756151/2661120*t^4 + 37869/640*t^3 + 26951489/1051050*t^2 + 120719/17160*t + 1

14172199/6227020800*t^14 + 839521/14826240*t^13 + 4345531/6842880*t^12 + 9693809/2280960*t^11 + 29466827/1555200*t^10 + 12346939/207360*t^9 + 593155051/4354560*t^8 + 335428619/1451520*t^7 + 1824926437/6220800*t^6 + 11561527/41472*t^5 + 672384883/3421440*t^4 + 57344251/570240*t^3 + 774504587/21621600*t^2 + 1481341/180180*t + 1

102869687/3487131648*t^14 + 93762997/166053888*t^13 + 26814797/5474304*t^12 + 5183975/202752*t^11 + 313285387/3483648*t^10 + 37514059/165888*t^9 + 10257579173/24385536*t^8 + 685133401/1161216*t^7 + 156630283/248832*t^6 + 7056737/13824*t^5 + 1491723449/4790016*t^4 + 7964861/57024*t^3 + 267071513/6054048*t^2 + 653483/72072*t + 1

1427029207/9686476800*t^14 + 642967681/276756480*t^13 + 17069149/1013760*t^12 + 226664183/3041280*t^11 + 2183411449/9676800*t^10 + 45647401/92160*t^9 + 11037166199/13547520*t^8 + 1978361233/1935360*t^7 + 113536009/115200*t^6 + 12594131/17280*t^5 + 543799349/1330560*t^4 + 1194569/7040*t^3 + 839599363/16816800*t^2 + 28867/3003*t + 1

2330931341/7264857600*t^14 + 52511021/11531520*t^13 + 24030767/798336*t^12 + 5839747/47520*t^11 + 1256846471/3628800*t^10 + 456531/640*t^9 + 562927961/508032*t^8 + 1267493/960*t^7 + 8815476293/7257600*t^6 + 29744423/34560*t^5 + 185254967/399168*t^4 + 5888683/31680*t^3 + 2669409121/50450400*t^2 + 237371/24024*t + 1

2047/81729648000*t^15 + 1637/1089728640*t^14 + 96113/2335132800*t^13 + 8171/11975040*t^12 + 6861559/898128000*t^11 + 332363/5443200*t^10 + 41021891/114307200*t^9 + 11994461/7620480*t^8 + 52777187/10206000*t^7 + 8612633/680400*t^6 + 127721333/5613300*t^5 + 21939781/748440*t^4 + 9890467843/378378000*t^3 + 291201101/18918900*t^2 + 200713/36036*t + 1

13824739/1307674368000*t^15 + 4553683/10897286400*t^14 + 6386407/849139200*t^13 + 23261/285120*t^12 + 4274811179/7185024000*t^11 + 16771973/5443200*t^10 + 5337022559/457228800*t^9 + 125133221/3810240*t^8 + 90183224827/1306368000*t^7 + 18717373/172800*t^6 + 45279007097/359251200*t^5 + 160705799/1496880*t^4 + 53994658409/825552000*t^3 + 2067214613/75675600*t^2 + 290287/40040*t + 1

26502841/40864824000*t^15 + 2029481/108972864*t^14 + 283464119/1167566400*t^13 + 542551/285120*t^12 + 4497391537/449064000*t^11 + 101954291/2721600*t^10 + 5879792693/57153600*t^9 + 805678021/3810240*t^8 + 419392199/1275750*t^7 + 2095357/5400*t^6 + 1945534273/5613300*t^5 + 86651311/374220*t^4 + 1781677097/15765750*t^3 + 182499931/4729725*t^2 + 767549/90090*t + 1

15041229521/1307674368000*t^15 + 2770152961/10897286400*t^14 + 23879655967/9340531200*t^13 + 13283921/855360*t^12 + 458466122761/7185024000*t^11 + 204880667/1088640*t^10 + 188694158389/457228800*t^9 + 2618336783/3810240*t^8 + 1148229601433/1306368000*t^7 + 1347973351/1555200*t^6 + 235811787067/359251200*t^5 + 141056449/374220*t^4 + 1454152371601/9081072000*t^3 + 3643452523/75675600*t^2 + 3407689/360360*t + 1

6423562433/81729648000*t^15 + 7707502519/5448643200*t^14 + 27326623759/2335132800*t^13 + 101215987/1710720*t^12 + 184063550921/898128000*t^11 + 936931991/1814400*t^10 + 112018446013/114307200*t^9 + 10893779219/7620480*t^8 + 16532849413/10206000*t^7 + 138967777/97200*t^6 + 5489557519/5613300*t^5 + 127591193/249480*t^4 + 226264391791/1135134000*t^3 + 349085689/6306300*t^2 + 17335/1716*t + 1

103795866137/435891456000*t^15 + 2720962597/726485760*t^14 + 85436922007/3113510400*t^13 + 248258957/1995840*t^12 + 935412781537/2395008000*t^11 + 326853251/362880*t^10 + 240462680389/152409600*t^9 + 2717930827/1270080*t^8 + 141164035703/62208000*t^7 + 1367854997/725760*t^6 + 146001312607/119750400*t^5 + 604375271/997920*t^4 + 2044084141171/9081072000*t^3 + 301806581/5045040*t^2 + 755383/72072*t + 1

2330931341/6810804000*t^15 + 2330931341/454053600*t^14 + 7024858627/194594400*t^13 + 52511021/332640*t^12 + 35934811877/74844000*t^11 + 18079861/16800*t^10 + 17502014809/9525600*t^9 + 1546240513/635040*t^8 + 537483109/212625*t^7 + 4324333/2100*t^6 + 1224216929/935550*t^5 + 13293361/20790*t^4 + 16605427634/70945875*t^3 + 96698018/1576575*t^2 + 95549/9009*t + 1

65519/20922789888000*t^16 + 79543/373621248000*t^15 + 6957607/1046139494400*t^14 + 337739/2668723200*t^13 + 188453959/114960384000*t^12 + 219851021/14370048000*t^11 + 31072651/292626432*t^10 + 72838013/130636800*t^9 + 326400589169/146313216000*t^8 + 17752960753/2612736000*t^7 + 89781614497/5748019200*t^6 + 19146782933/718502400*t^5 + 4776994588679/145297152000*t^4 + 10484040029/370656000*t^3 + 3261990821/201801600*t^2 + 4107353/720720*t + 1

8386549/4184557977600*t^16 + 235577347/2615348736000*t^15 + 1941159433/1046139494400*t^14 + 433619663/18681062400*t^13 + 4527594989/22992076800*t^12 + 2460092321/2052864000*t^11 + 7895990153/1463132160*t^10 + 16726811921/914457600*t^9 + 1376054702539/29262643200*t^8 + 239810173651/2612736000*t^7 + 778439638303/5748019200*t^6 + 15362179829/102643200*t^5 + 10638261256487/87178291200*t^4 + 1300144121167/18162144000*t^3 + 17510765017/605404800*t^2 + 1789913/240240*t + 1

238139017/1394852659200*t^16 + 4884188689/871782912000*t^15 + 2663021831/31701196800*t^14 + 4748941721/6227020800*t^13 + 5128927319/1094860800*t^12 + 98751284309/4790016000*t^11 + 163819203937/2438553600*t^10 + 50389134767/304819200*t^9 + 3030635875639/9754214400*t^8 + 390559324657/870912000*t^7 + 27211313177/54743040*t^6 + 100738637621/239500800*t^5 + 28323235169/105670656*t^4 + 2278299817967/18162144000*t^3 + 924827489/22422400*t^2 + 1268081/144144*t + 1

12197605507/2988969984000*t^16 + 38588349623/373621248000*t^15 + 178582288211/149448499200*t^14 + 22385336959/2668723200*t^13 + 656910348587/16422912000*t^12 + 282339434323/2052864000*t^11 + 369224306399/1045094400*t^10 + 90549962953/130636800*t^9 + 22000489914877/20901888000*t^8 + 3253182966113/2612736000*t^7 + 942439665749/821145600*t^6 + 84079919659/102643200*t^5 + 27785051592401/62270208000*t^4 + 5789518963/32032000*t^3 + 4500237263/86486400*t^2 + 7075451/720720*t + 1

782115518299/20922789888000*t^16 + 2002909883633/2615348736000*t^15 + 302152547267/41845579776*t^14 + 778256010661/18681062400*t^13 + 2708975220197/16422912000*t^12 + 6844212677053/14370048000*t^11 + 7606317001943/7315660800*t^10 + 1604326049707/914457600*t^9 + 339126687662629/146313216000*t^8 + 6300404941409/2612736000*t^7 + 1621382418443/821145600*t^6 + 907747514761/718502400*t^5 + 90389584514179/145297152000*t^4 + 4168694454773/18162144000*t^3 + 12230913041/201801600*t^2 + 2535431/240240*t + 1

3207483178157/20922789888000*t^16 + 7040798154763/2615348736000*t^15 + 23024171822341/1046139494400*t^14 + 2080712000639/18681062400*t^13 + 44966635121557/114960384000*t^12 + 2077812166649/2052864000*t^11 + 14623375709737/7315660800*t^10 + 2814160247513/914457600*t^9 + 546892811590067/146313216000*t^8 + 1343506410637/373248000*t^7 + 3155289994583/1149603840*t^6 + 168771140057/102643200*t^5 + 110678531004277/145297152000*t^4 + 1607840669861/6054048000*t^3 + 13427417207/201801600*t^2 + 1590299/144144*t + 1

12157712239/39852933120*t^16 + 4300583023337/871782912000*t^15 + 13073198753273/348713164800*t^14 + 1107071012233/6227020800*t^13 + 4514243934229/7664025600*t^12 + 6927320056717/4790016000*t^11 + 135532864157/49766400*t^10 + 1223814858151/304819200*t^9 + 45716446614323/9754214400*t^8 + 3790096323401/870912000*t^7 + 6147882828239/1916006400*t^6 + 446280748393/239500800*t^5 + 3481763779513/4151347200*t^4 + 5170327419391/18162144000*t^3 + 2807586737/40360320*t^2 + 1624333/144144*t + 1

851/2309658624000*t^17 + 1/35481600*t^16 + 78907/79252992000*t^15 + 156553/7264857600*t^14 + 79614641/249080832000*t^13 + 109939/31933440*t^12 + 799940987/28740096000*t^11 + 249883/1451520*t^10 + 20056930547/24385536000*t^9 + 618304501/203212800*t^8 + 3779131133/435456000*t^7 + 753599443/39916800*t^6 + 37302932341/1213056000*t^5 + 26014811/712800*t^4 + 138029148779/4540536000*t^3 + 284302033/16816800*t^2 + 35655743/6126120*t + 1

704339/1976041267200*t^17 + 68179/3773952000*t^16 + 61326743/145297152000*t^15 + 3980737/660441600*t^14 + 488296163/8302694400*t^13 + 60269063/145152000*t^12 + 24488274067/11176704000*t^11 + 7985869/907200*t^10 + 4419148049/162570240*t^9 + 131484174893/2032128000*t^8 + 47274030061/399168000*t^7 + 60078493/362880*t^6 + 141092116799/807206400*t^5 + 3591628709/26208000*t^4 + 13067695063/168168000*t^3 + 7763713/254800*t^2 + 1949768/255255*t + 1

297507989/7113748561920*t^17 + 1807009/1162377216*t^16 + 2104789937/79252992000*t^15 + 2008583779/7264857600*t^14 + 19539472819/9963233280*t^13 + 1603482653/159667200*t^12 + 7723180761799/201180672000*t^11 + 5678848963/50803200*t^10 + 244750541353/975421440*t^9 + 88794772351/203212800*t^8 + 2832667066253/4790016000*t^7 + 4952629601/7983360*t^6 + 1189860677519/2377589760*t^5 + 3856498909/12612600*t^4 + 625827521629/4540536000*t^3 + 2210167433/50450400*t^2 + 55482673/6126120*t + 1

23667665053/17784371404800*t^17 + 22238830951/581188608000*t^16 + 220257696289/435891456000*t^15 + 118207941083/29059430400*t^14 + 50612988793/2264371200*t^13 + 3690293683/41472000*t^12 + 26752600927123/100590336000*t^11 + 124043364959/203212800*t^10 + 2661179006849/2438553600*t^9 + 6227302841161/4064256000*t^8 + 2029147364353/1197504000*t^7 + 33592602803/22809600*t^6 + 65241406255889/65383718400*t^5 + 6281184766093/12108096000*t^4 + 83147049481/412776000*t^3 + 255870553/4586400*t^2 + 2221493/218790*t + 1

317627331799/19760412672000*t^17 + 54218672303/145297152000*t^16 + 1161028375763/290594304000*t^15 + 190552702177/7264857600*t^14 + 9831588738211/83026944000*t^13 + 44569374589/114048000*t^12 + 21865147646677/22353408000*t^11 + 96557160337/50803200*t^10 + 23643441029657/8128512000*t^9 + 3591016172633/1016064000*t^8 + 5457250742837/1596672000*t^7 + 14958668713/5702400*t^6 + 12784002957967/8072064000*t^5 + 93311811071/126126000*t^4 + 43742812109/168168000*t^3 + 73652279/1121120*t^2 + 22408157/2042040*t + 1

7763913237097/88921857024000*t^17 + 166233279019/96864768000*t^16 + 1368643688767/87178291200*t^15 + 646075860137/7264857600*t^14 + 43585058599091/124540416000*t^13 + 1623727534987/1596672000*t^12 + 9105264795677/4023613440*t^11 + 50100215143/12700800*t^10 + 66586642409477/12192768000*t^9 + 12291649664027/2032128000*t^8 + 183700501363/34214400*t^7 + 75963242323/19958400*t^6 + 697790875792469/326918592000*t^5 + 16908191971903/18162144000*t^4 + 278824725719/908107200*t^3 + 368595853/5045040*t^2 + 8839451/765765*t + 1

8313722318537/35568742809600*t^17 + 599392631477/145297152000*t^16 + 4263051636677/124540416000*t^15 + 429823169297/2421619200*t^14 + 32111958139951/49816166400*t^13 + 51482589931/29568000*t^12 + 729176868534623/201180672000*t^11 + 60389100799/10160640*t^10 + 5419775952331/696729600*t^9 + 2770725454369/338688000*t^8 + 33156078888181/4790016000*t^7 + 20816736929/4435200*t^6 + 330259496380969/130767436800*t^5 + 2406274480067/2270268000*t^4 + 218735303519/648648000*t^3 + 3914455513/50450400*t^2 + 14554933/1225224*t + 1

12157712239/37638881280*t^17 + 12157712239/2214051840*t^16 + 4270327196879/96864768000*t^15 + 4300583023337/19372953600*t^14 + 869318708893/1107025920*t^13 + 881922464629/425779200*t^12 + 4491616788193/1064448000*t^11 + 131300068667/19353600*t^10 + 945083142599/108380160*t^9 + 4887309849203/541900800*t^8 + 1999600942483/266112000*t^7 + 106926417811/21288960*t^6 + 13667854873/5125120*t^5 + 28325682897/25625600*t^4 + 38971831489/112112000*t^3 + 887003237/11211200*t^2 + 1632341/136136*t + 1

233/5690998849536*t^18 + 166003/47424990412800*t^17 + 1452379/10461394944000*t^16 + 11824403/3487131648000*t^15 + 238882003/4184557977600*t^14 + 19999913/28466380800*t^13 + 1754618897/268240896000*t^12 + 4204520279/89413632000*t^11 + 3090581059/11705057280*t^10 + 4540153721/3901685760*t^9 + 809023899161/201180672000*t^8 + 207449407397/19160064000*t^7 + 11731108027757/523069747200*t^6 + 6110353614787/174356582400*t^5 + 4376505156521/108972864000*t^4 + 589936188311/18162144000*t^3 + 11333378761/643242600*t^2 + 36356443/6126120*t + 1

95609981/1600593426432000*t^18 + 1205664703/355687428096000*t^17 + 5593238411/62768369664000*t^16 + 278956879/193729536000*t^15 + 31387738427/1961511552000*t^14 + 64575712129/498161664000*t^13 + 521214745361/658409472000*t^12 + 1495782924197/402361344000*t^11 + 2978815143391/219469824000*t^10 + 629235627001/16257024000*t^9 + 416070184625231/4828336128000*t^8 + 714050321587/4790016000*t^7 + 4675566296705053/23538138624000*t^6 + 263232550979771/1307674368000*t^5 + 199224188101073/1307674368000*t^4 + 507097796243/6054048000*t^3 + 89705061133/2806876800*t^2 + 95756029/12252240*t + 1

614033131/64023737057280*t^18 + 28504142893/71137485619200*t^17 + 485453440957/62768369664000*t^16 + 159531071857/1743565824000*t^15 + 145595591359/196151155200*t^14 + 436145564243/99632332800*t^13 + 140644874268077/7242504192000*t^12 + 26620715360299/402361344000*t^11 + 1538992167497/8778792960*t^10 + 3548631006449/9754214400*t^9 + 2862991609635097/4828336128000*t^8 + 3632817779839/4790016000*t^7 + 3566995815971047/4707627724800*t^6 + 153179633712929/261534873600*t^5 + 450597042770551/1307674368000*t^4 + 2727132028483/18162144000*t^3 + 1428986729017/30875644800*t^2 + 37983109/4084080*t + 1

59606993/148203095040*t^18 + 103070503441/7904165068800*t^17 + 123675281213/634023936000*t^16 + 1038723191939/581188608000*t^15 + 3916692396311/348713164800*t^14 + 341368689793/6642155520*t^13 + 15867325735147/89413632000*t^12 + 7049520334817/14902272000*t^11 + 2409293068859/2438553600*t^10 + 589469382023/361267200*t^9 + 1148425548893003/536481792000*t^8 + 3563661993013/1596672000*t^7 + 29194056434969/15850598400*t^6 + 34625380358461/29059430400*t^5 + 86336323100149/145297152000*t^4 + 4034674669163/18162144000*t^3 + 67948416037/1143542400*t^2 + 128262749/12252240*t + 1

722452584787/114328101888000*t^18 + 1200227144851/7258927104000*t^17 + 3588174605551/1793381990400*t^16 + 741434923813/49816166400*t^15 + 171108186103189/2241727488000*t^14 + 20418150051011/71165952000*t^13 + 24251313867581/29561241600*t^12 + 21011031126823/11496038400*t^11 + 25257154622101/7838208000*t^10 + 31624378151897/6967296000*t^9 + 707952434563243/137952460800*t^8 + 892396541171/191600640*t^7 + 11357703030011257/3362591232000*t^6 + 361941482175569/186810624000*t^5 + 32274549272437/37362124800*t^4 + 16776186971/57657600*t^3 + 8900240737/126023040*t^2 + 139250563/12252240*t + 1

2859970743073/64023737057280*t^18 + 69868595349463/71137485619200*t^17 + 631133766441751/62768369664000*t^16 + 37067642421937/581188608000*t^15 + 55181543659849/196151155200*t^14 + 91440463582057/99632332800*t^13 + 16647977649915431/7242504192000*t^12 + 1821170204284657/402361344000*t^11 + 62452772949563/8778792960*t^10 + 5856473488877/650280960*t^9 + 44518440441850411/4828336128000*t^8 + 5221110430751/684288000*t^7 + 23901862899412837/4707627724800*t^6 + 702993171640787/261534873600*t^5 + 1454099002165573/1307674368000*t^4 + 6342720401089/18162144000*t^3 + 2451483910867/30875644800*t^2 + 49084619/4084080*t + 1

27964165775423/177843714048000*t^18 + 361610774423567/118562476032000*t^17 + 581727322008907/20922789888000*t^16 + 276289318543541/1743565824000*t^15 + 414139207866323/653837184000*t^14 + 940211093659603/498161664000*t^13 + 1166473386739363/268240896000*t^12 + 354459078574343/44706816000*t^11 + 849086088021599/73156608000*t^10 + 670948693389041/48771072000*t^9 + 21324538182730127/1609445376000*t^8 + 7088393014051/684288000*t^7 + 17097925347230639/2615348736000*t^6 + 1435824692425259/435891456000*t^5 + 567118716450961/435891456000*t^4 + 7106317213379/18162144000*t^3 + 175815722539/2058376320*t^2 + 152485811/12252240*t + 1

37307713155613/128047474114560*t^18 + 47031555912931/8892185702400*t^17 + 407436979196483/8966909952000*t^16 + 214018478206613/871782912000*t^15 + 5868601005997459/6276836966400*t^14 + 3689230964611/1383782400*t^13 + 42735801871920691/7242504192000*t^12 + 1042982805726023/100590336000*t^11 + 26284556740249/1791590400*t^10 + 4109330500103/243855360*t^9 + 76111241670445301/4828336128000*t^8 + 38301948776473/3193344000*t^7 + 6949001216883877/941525544960*t^6 + 237461864531083/65383718400*t^5 + 262103113599269/186810624000*t^4 + 2501882491093/6054048000*t^3 + 2731594237649/30875644800*t^2 + 31014479/2450448*t + 1

14563/3379030566912000*t^19 + 131063/320118685286400*t^18 + 751/41455411200*t^17 + 778069/1569209241600*t^16 + 49179223/5230697472000*t^15 + 411405271/3138418483200*t^14 + 364016971/261534873600*t^13 + 4164429253/362125209600*t^12 + 60275944877/804722688000*t^11 + 16994366707/43893964800*t^10 + 128003848537/80472268800*t^9 + 312489783167/60354201600*t^8 + 131198061131/9906624000*t^7 + 6179696717027/235381386240*t^6 + 198738510961/5029516800*t^5 + 716983186781/16345929600*t^4 + 1332758314439/38594556000*t^3 + 141306446467/7718911200*t^2 + 234454999/38798760*t + 1

383925299/40548366802944000*t^19 + 382702049/640237370572800*t^18 + 2498165393/142274971238400*t^17 + 799958843/2510734786560*t^16 + 83658827363/20922789888000*t^15 + 3554999807/96566722560*t^14 + 293772652217/1141243084800*t^13 + 80880857693/57940033536*t^12 + 19132646548067/3218890752000*t^11 + 1755294170737/87787929600*t^10 + 34254300039409/643778150400*t^9 + 21572558614067/193133445120*t^8 + 2883742645508111/15692092416000*t^7 + 1104291710997947/4707627724800*t^6 + 119827282380061/523069747200*t^5 + 8782525499477/52306974720*t^4 + 139952316011/1559376000*t^3 + 79317201431/2375049600*t^2 + 1858243391/232792560*t + 1

17657989/8532905472000*t^19 + 321800797/3334569638400*t^18 + 9287523041/4446092851200*t^17 + 33730871/1210809600*t^16 + 335063338991/1307674368000*t^15 + 10248451363/5943974400*t^14 + 574597030699/65383718400*t^13 + 47548119343/1371686400*t^12 + 21570912362249/201180672000*t^11 + 9985859089/38102400*t^10 + 929132651833/1828915200*t^9 + 3929027998339/5029516800*t^8 + 77666412630511/81729648000*t^7 + 278538047807/306486180*t^6 + 1227755356853/1816214400*t^5 + 1762678621/4586400*t^4 + 3133818853631/19297278000*t^3 + 2846174749/58476600*t^2 + 554616397/58198140*t + 1

9197440123/81096733605888*t^19 + 26364254107/6402373705728*t^18 + 492860313037/7113748561920*t^17 + 40927619693/57062154240*t^16 + 214142429687/41845579776*t^15 + 128999803061/4828336128*t^14 + 66408327281423/627683696640*t^13 + 23550207025073/72425041920*t^12 + 563264314355/715309056*t^11 + 1331928453203/877879296*t^10 + 75291756329821/32188907520*t^9 + 139606122066463/48283361280*t^8 + 44902726121927/15692092416*t^7 + 9649216058915/4279661568*t^6 + 36599807557249/26153487360*t^5 + 8790062108029/13076743680*t^4 + 299910422587/1235025792*t^3 + 1494689587/23750496*t^2 + 27843779/2586584*t + 1

829745172187/362038989312000*t^19 + 5379045855709/80029671321600*t^18 + 369570774673/404190259200*t^17 + 273513985309/35663846400*t^16 + 58162032029483/1307674368000*t^15 + 148771360455461/784604620800*t^14 + 17283559731803/28021593600*t^13 + 12909931332793/8230118400*t^12 + 212576326171499/67060224000*t^11 + 11290082116237/2194698240*t^10 + 135697476118529/20118067200*t^9 + 108052481576887/15088550400*t^8 + 19570737871397/3184272000*t^7 + 113377254791239/26747884800*t^6 + 37972515708161/16345929600*t^5 + 4059566408461/4086482400*t^4 + 1035530065133/3216213000*t^3 + 13227990703/175429800*t^2 + 341417077/29099070*t + 1

10031448420143/482718652416000*t^19 + 27204534921281/53353114214400*t^18 + 207426123806621/35568742809600*t^17 + 14425623415711/348713164800*t^16 + 1068670174657951/5230697472000*t^15 + 390964132068203/523069747200*t^14 + 104796223253713/49816166400*t^13 + 563483373196043/120708403200*t^12 + 6679856853039019/804722688000*t^11 + 9714128247953/812851200*t^10 + 2254880845145933/160944537600*t^9 + 1078150406831453/80472268800*t^8 + 243509180638649/23351328000*t^7 + 2575037132913251/392302310400*t^6 + 5331730992737/1614412800*t^5 + 1052276154557/807206400*t^4 + 60509242143277/154378224000*t^3 + 220138221781/2572970400*t^2 + 1450653161/116396280*t + 1

21753114839033/230388447744000*t^19 + 20198547534727/10003708915200*t^18 + 12884729492009/635156121600*t^17 + 12499282684429/98075577600*t^16 + 104984003221543/186810624000*t^15 + 362908890352681/196151155200*t^14 + 84184471911901/17831923200*t^13 + 432935777242121/45265651200*t^12 + 449124904895377/28740096000*t^11 + 570488884345/27433728*t^10 + 5919271364417/261273600*t^9 + 305281217653463/15088550400*t^8 + 3622160399158963/245188944000*t^7 + 161068297877521/18389170800*t^6 + 1389955462099/333590400*t^5 + 6376306892609/4086482400*t^4 + 111901277333/250614000*t^3 + 179541754627/1929727800*t^2 + 83892023/6466460*t + 1

515968158649877/2252687044608000*t^19 + 1436047812847939/320118685286400*t^18 + 46973308288889/1129166438400*t^17 + 1524733275877817/6276836966400*t^16 + 1497195450161443/1494484992000*t^15 + 9734759838965477/3138418483200*t^14 + 15650916473755991/2092278988800*t^13 + 10418559801910147/724250419200*t^12 + 5149807033456907/229920768000*t^11 + 1251070570117283/43893964800*t^10 + 1369716231755393/45984153600*t^9 + 12355278404969777/482833612800*t^8 + 47124923181428129/2615348736000*t^7 + 24306463927540129/2353813862400*t^6 + 25396639649291/5337446400*t^5 + 226129677717071/130767436800*t^4 + 3536577750991/7351344000*t^3 + 1506987616129/15437822400*t^2 + 772605221/58198140*t + 1

10:

37307713155613/121645100408832*t^19 + 37307713155613/6402373705728*t^18 + 280601677603927/5335311421440*t^17 + 47031555912931/156920924160*t^16 + 75892347274115/62768369664*t^15 + 230298830387741/62768369664*t^14 + 2044702971739469/235381386240*t^13 + 594781299506467/36212520960*t^12 + 243121543492657/9656672256*t^11 + 27720120825521/877879296*t^10 + 785985997700501/24141680640*t^9 + 166695460499873/6035420160*t^8 + 226211942760709/11769069312*t^7 + 256440058580645/23538138624*t^6 + 32484739561843/6538371840*t^5 + 2922832026659/1634592960*t^4 + 76094572289/154378224*t^3 + 15307582769/154378224*t^2 + 155685007/11639628*t + 1

Definition

The hypersimplex $\Delta(k,n)$ is the lattice polytope $\{x\in[0,1]^n:x_1+\cdots+x_n=k\}$, equivalently the convex hull of the $0$-$1$ vectors in $\mathbb{R}^n$ with exactly $k$ ones [3]. The table stores its Ehrhart polynomial $L_{\Delta(k,n)}(t)$ [4].

Parameters

$n$

— ambient dimension ($n$ is an integer with $n\geq4$)

$k$

— coordinate sum ($k$ is an integer with $2\leq k\leq\lfloor n/2\rfloor$)

Formulas

(1)

$L_{\Delta(k,n)}(t)$ is the coefficient of $q^{kt}$ in $(1+q+\cdots+q^t)^n$.

(2)

$L_{\Delta(k,n)}(t)=\sum_{i=0}^{k-1}(-1)^i\binom{n}{i} \binom{(k-i)t-i+n-1}{n-1}$.

(3)

$\sum_{t\geq0}L_{\Delta(k,n)}(t)z^t= h^*_{\Delta(k,n)}(z)/(1-z)^n$.

(4)

$h^*_{\Delta(k,n)}(1)$ is the Eulerian number $A(n-1,k-1)$, the number of permutations of $n-1$ letters with $k-1$ descents, and it is the normalized volume of $\Delta(k,n)$.

Comments

(5)

The hypersimplex $\Delta(k,n)$ is the matroid polytope of the uniform matroid $U_{k,n}$ [3] and the moment polytope for the torus action on the Grassmannian $\mathrm{Gr}(k,n)$ [1]. The coefficient formula (1) says that $L_{\Delta(k,n)}(t)$ counts the ways $n$ dice with faces $0,1,\dots,t$ can sum to $kt$. It is also the Hilbert function of the squarefree Veronese algebra generated by the degree-$k$ squarefree monomials in $n$ variables [2].

(6)

Lattice points are counted in $\mathbb{Z}^n$. Since $t\Delta(k,n)$ lies in the hyperplane $x_1+\cdots+x_n=kt$, the normalized volume in (4) is taken with respect to the lattice $\{x\in\mathbb{Z}^n:x_1+\cdots+x_n=0\}$.

(7)

The map $x\mapsto\mathbf{1}-x$ identifies $\Delta(k,n)$ with $\Delta(n-k,n)$, so $\Delta(n-k,n)$ has the same two polynomials; for $k>n/2$, look up $\Delta(n-k,n)$.

(8)

The omitted case $k=1$ is the standard simplex, with $L(t)=\binom{t+n-1}{n-1}$.

Programs

(P1)

Sage

n, k = 6, 2
T = PolynomialRing(QQ, "t"); t = T.gen()

sum((-1)**i * binomial(n, i) * binomial((k - i)*t - i + n - 1, n - 1)
    for i in range(k))

References

[1]

Alexander Postnikov, Positive Grassmannian and polyhedral subdivisions, 2018. (arXiv)

[2]

Mordechai Katzman, The Hilbert series of algebras of Veronese type, Communications in Algebra 33 (2005), no. 4, 1141-1146. (arXiv) (doi)

Links

[3]

Wikipedia: Hypersimplex

[4]

Wikipedia: Ehrhart polynomial

Similar tables

Ehrhart $h^*$-polynomials of the hypersimplices — gives the numerator $h^*_{\Delta(k,n)}(z)$ of the Ehrhart series; this table gives the counting polynomial $L_{\Delta(k,n)}(t)$, and the two determine each other by (3). A reader holding an Ehrhart counting polynomial wants this table; a reader holding the numerator wants T244.

Ehrhart polynomials of the Birkhoff polytopes — the transportation-polytope Ehrhart-polynomial analogue with all row and column sums fixed

Data properties

Entries are of type: rational polynomial

How well the digits are known: exact

Table is complete: no (it holds $L_{\Delta(k,n)}(t)$ for every pair with $4\leq n\leq20$ and $2\leq k\leq\lfloor n/2\rfloor$; the range stops at $n=20$ because the longest stored polynomial, $L_{\Delta(9,20)}(t)$, has 682 characters)

How they were obtained:

The Ehrhart polynomials are computed exactly from (2), an inclusion-exclusion count of bounded integer compositions.

The rows were checked by direct coefficient counts from (1); their normalized volumes were checked against Eulerian numbers through (4); and the symmetry $\Delta(k,n)\cong\Delta(n-k,n)$ was checked on the computed polynomials.