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Ehrhart $h^*$-polynomials of the hypersimplices
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Polynomials
$n$
$k$ 
$h^*_{\Delta(k,n)}(z)$
4
2:
z^2 + 2*z + 1
5
2:
5*z^2 + 5*z + 1
6
2:
z^3 + 15*z^2 + 9*z + 1
6
3:
z^4 + 14*z^3 + 36*z^2 + 14*z + 1
7
2:
7*z^3 + 35*z^2 + 14*z + 1
7
3:
21*z^4 + 119*z^3 + 133*z^2 + 28*z + 1
8
2:
z^4 + 28*z^3 + 70*z^2 + 20*z + 1
8
3:
8*z^5 + 202*z^4 + 568*z^3 + 364*z^2 + 48*z + 1
8
4:
z^6 + 62*z^5 + 575*z^4 + 1140*z^3 + 575*z^2 + 62*z + 1
9
2:
9*z^4 + 84*z^3 + 126*z^2 + 27*z + 1
9
3:
z^6 + 147*z^5 + 1230*z^4 + 2005*z^3 + 834*z^2 + 75*z + 1
9
4:
84*z^6 + 1548*z^5 + 5895*z^4 + 6165*z^3 + 1809*z^2 + 117*z + 1
10
2:
z^5 + 45*z^4 + 210*z^3 + 210*z^2 + 35*z + 1
10
3:
55*z^6 + 1352*z^5 + 5565*z^4 + 5830*z^3 + 1695*z^2 + 110*z + 1
10
4:
45*z^7 + 2400*z^6 + 18328*z^5 + 37985*z^4 + 24565*z^3 + 4710*z^2 + 200*z + 1
10
5:
z^8 + 242*z^7 + 6478*z^6 + 37994*z^5 + 66760*z^4 + 37994*z^3 + 6478*z^2 + 242*z + 1
11
2:
11*z^5 + 165*z^4 + 462*z^3 + 330*z^2 + 44*z + 1
11
3:
11*z^7 + 869*z^6 + 8437*z^5 + 20438*z^4 + 14773*z^3 + 3157*z^2 + 154*z + 1
11
4:
11*z^8 + 2156*z^7 + 35508*z^6 + 142428*z^5 + 183711*z^4 + 80278*z^3 + 10780*z^2 + 319*z + 1
11
5:
330*z^8 + 16225*z^7 + 158235*z^6 + 466004*z^5 + 478236*z^4 + 171831*z^3 + 19041*z^2 + 451*z + 1
12
2:
z^6 + 66*z^5 + 495*z^4 + 924*z^3 + 495*z^2 + 54*z + 1
12
3:
z^8 + 352*z^7 + 7930*z^6 + 40612*z^5 + 64285*z^4 + 33748*z^3 + 5500*z^2 + 208*z + 1
12
4:
z^9 + 1209*z^8 + 44682*z^7 + 344488*z^6 + 834900*z^5 + 727638*z^4 + 227646*z^3 + 22440*z^2 + 483*z + 1
12
5:
220*z^9 + 25674*z^8 + 441024*z^7 + 2176954*z^6 + 3836196*z^5 + 2574429*z^4 + 633996*z^3 + 48840*z^2 + 780*z + 1
12
6:
z^10 + 912*z^9 + 62767*z^8 + 878932*z^7 + 3834832*z^6 + 6169360*z^5 + 3834832*z^4 + 878932*z^3 + 62767*z^2 + 912*z + 1
13
2:
13*z^6 + 286*z^5 + 1287*z^4 + 1716*z^3 + 715*z^2 + 65*z + 1
13
3:
91*z^8 + 4992*z^7 + 51610*z^6 + 161811*z^5 + 179335*z^4 + 71071*z^3 + 9087*z^2 + 273*z + 1
13
4:
442*z^9 + 38727*z^8 + 568087*z^7 + 2487862*z^6 + 3985813*z^5 + 2483052*z^4 + 579592*z^3 + 43407*z^2 + 702*z + 1
13
5:
78*z^10 + 26728*z^9 + 853658*z^8 + 7080853*z^7 + 20751926*z^6 + 24129937*z^5 + 11341720*z^4 + 2019108*z^3 + 113191*z^2 + 1274*z + 1
13
6:
1287*z^10 + 154375*z^9 + 3334110*z^8 + 21984040*z^7 + 54615067*z^6 + 55490487*z^5 + 23102326*z^4 + 3649477*z^3 + 179413*z^2 + 1703*z + 1
14
2:
z^7 + 91*z^6 + 1001*z^5 + 3003*z^4 + 3003*z^3 + 1001*z^2 + 77*z + 1
14
3:
14*z^9 + 2184*z^8 + 45278*z^7 + 265174*z^6 + 557662*z^5 + 454545*z^4 + 140140*z^3 + 14378*z^2 + 350*z + 1
14
4:
105*z^10 + 24192*z^9 + 674492*z^8 + 5177147*z^7 + 14424501*z^6 + 16255043*z^5 + 7542731*z^4 + 1355081*z^3 + 79170*z^2 + 987*z + 1
14
5:
14*z^11 + 19187*z^10 + 1191736*z^9 + 16675036*z^8 + 79494076*z^7 + 152350576*z^6 + 124579364*z^5 + 42982562*z^4 + 5744662*z^3 + 242333*z^2 + 1988*z + 1
14
6:
1001*z^11 + 256256*z^10 + 9019297*z^9 + 91077427*z^8 + 344909878*z^7 + 550074602*z^6 + 383162430*z^5 + 113644440*z^4 + 13012545*z^3 + 460642*z^2 + 2989*z + 1
14
7:
z^12 + 3418*z^11 + 568270*z^10 + 16972890*z^9 + 155600485*z^8 + 549831736*z^7 + 829218404*z^6 + 549831736*z^5 + 155600485*z^4 + 16972890*z^3 + 568270*z^2 + 3418*z + 1
15
2:
15*z^7 + 455*z^6 + 3003*z^5 + 6435*z^4 + 5005*z^3 + 1365*z^2 + 90*z + 1
15
3:
z^10 + 665*z^9 + 28650*z^8 + 305160*z^7 + 1141140*z^6 + 1712532*z^5 + 1065090*z^4 + 261690*z^3 + 21945*z^2 + 440*z + 1
15
4:
15*z^11 + 11178*z^10 + 601820*z^9 + 7936815*z^8 + 36917520*z^7 + 70508540*z^6 + 58480587*z^5 + 20859765*z^4 + 2955615*z^3 + 137580*z^2 + 1350*z + 1
15
5:
z^12 + 9813*z^11 + 1245141*z^10 + 29505050*z^9 + 225079320*z^8 + 686074008*z^7 + 917619424*z^6 + 552064083*z^5 + 144716100*z^4 + 14939655*z^3 + 486591*z^2 + 2988*z + 1
15
6:
455*z^12 + 299910*z^11 + 18044510*z^10 + 282064700*z^9 + 1611895470*z^8 + 3916933725*z^7 + 4316832045*z^6 + 2175342072*z^5 + 479511015*z^4 + 41247325*z^3 + 1086645*z^2 + 4990*z + 1
15
7:
5005*z^12 + 1401960*z^11 + 61742562*z^10 + 806191155*z^9 + 4058378310*z^8 + 8918175720*z^7 + 9011739220*z^6 + 4191072999*z^5 + 853398330*z^4 + 67456320*z^3 + 1608090*z^2 + 6420*z + 1
16
2:
z^8 + 120*z^7 + 1820*z^6 + 8008*z^5 + 12870*z^4 + 8008*z^3 + 1820*z^2 + 104*z + 1
16
3:
136*z^10 + 13328*z^9 + 258314*z^8 + 1656496*z^7 + 4273272*z^6 + 4786112*z^5 + 2337296*z^4 + 466752*z^3 + 32488*z^2 + 544*z + 1
16
4:
z^12 + 3844*z^11 + 415618*z^10 + 9342268*z^9 + 70495535*z^8 + 217730376*z^7 + 300515548*z^6 + 189891192*z^5 + 53381039*z^4 + 6084116*z^3 + 229570*z^2 + 1804*z + 1
16
5:
3620*z^12 + 1003440*z^11 + 40556456*z^10 + 489024224*z^9 + 2308167620*z^8 + 4832050656*z^7 + 4726533952*z^6 + 2164248528*z^5 + 442618800*z^4 + 36091424*z^3 + 926448*z^2 + 4352*z + 1
16
6:
120*z^13 + 256650*z^12 + 27571400*z^11 + 673488900*z^10 + 5762482896*z^9 + 20861071757*z^8 + 34944643608*z^7 + 27990945724*z^6 + 10603216848*z^5 + 1791956640*z^4 + 118964064*z^3 + 2392328*z^2 + 7992*z + 1
16
7:
4368*z^13 + 2450448*z^12 + 167405664*z^11 + 3167382488*z^10 + 22698040432*z^9 + 71539991310*z^8 + 106556585360*z^7 + 76751967352*z^6 + 26260826176*z^5 + 4001548076*z^4 + 237225072*z^3 + 4160240*z^2 + 11424*z + 1
16
8:
z^14 + 12854*z^13 + 4990827*z^12 + 297458828*z^11 + 5202902329*z^10 + 35289282282*z^9 + 106517454171*z^8 + 152914614888*z^7 + 106517454171*z^6 + 35289282282*z^5 + 5202902329*z^4 + 297458828*z^3 + 4990827*z^2 + 12854*z + 1
17
2:
17*z^8 + 680*z^7 + 6188*z^6 + 19448*z^5 + 24310*z^4 + 12376*z^3 + 2380*z^2 + 119*z + 1
17
3:
17*z^11 + 4556*z^10 + 165087*z^9 + 1789114*z^8 + 7611172*z^7 + 14298547*z^6 + 12364882*z^5 + 4851392*z^4 + 800462*z^3 + 46852*z^2 + 663*z + 1
17
4:
969*z^12 + 226083*z^11 + 8700294*z^10 + 104604842*z^9 + 505219991*z^8 + 1102887104*z^7 + 1144219017*z^6 + 565976240*z^5 + 127954223*z^4 + 11924106*z^3 + 370022*z^2 + 2363*z + 1
17
5:
952*z^13 + 637058*z^12 + 44517050*z^11 + 841756071*z^10 + 6027751421*z^9 + 19111651396*z^8 + 28920688609*z^7 + 21433283664*z^6 + 7669449445*z^5 + 1249845287*z^4 + 81965058*z^3 + 1686366*z^2 + 6171*z + 1
17
6:
17*z^14 + 164815*z^13 + 33084788*z^12 + 1274618911*z^11 + 16231198545*z^10 + 86017327600*z^9 + 212508483425*z^8 + 258066480485*z^7 + 155899626505*z^6 + 45697307028*z^5 + 6064854533*z^4 + 317387535*z^3 + 4971752*z^2 + 12359*z + 1
17
7:
2380*z^14 + 3157852*z^13 + 349751591*z^12 + 9722297365*z^11 + 99053770273*z^10 + 441975864607*z^9 + 945669868474*z^8 + 1010629719085*z^7 + 541647432237*z^6 + 141103063485*z^5 + 16562470320*z^4 + 755762200*z^3 + 9998856*z^2 + 19431*z + 1
17
8:
19448*z^14 + 12447536*z^13 + 1063060592*z^12 + 25511378848*z^11 + 234802079884*z^10 + 968180708188*z^9 + 1938631187883*z^8 + 1952649084909*z^7 + 989768069669*z^6 + 243983890741*z^5 + 27027356245*z^4 + 1155521433*z^3 + 14095805*z^2 + 24293*z + 1
18
2:
z^9 + 153*z^8 + 3060*z^7 + 18564*z^6 + 43758*z^5 + 43758*z^4 + 18564*z^3 + 3060*z^2 + 135*z + 1
18
3:
z^12 + 1122*z^11 + 80580*z^10 + 1480784*z^9 + 10157058*z^8 + 30604896*z^7 + 43579977*z^6 + 29883960*z^5 + 9598914*z^4 + 1326884*z^3 + 66045*z^2 + 798*z + 1
18
4:
171*z^13 + 97545*z^12 + 6543504*z^11 + 124294599*z^10 + 917273982*z^9 + 3053143611*z^8 + 4929475482*z^7 + 3963093702*z^6 + 1568492523*z^5 + 289999719*z^4 + 22402770*z^3 + 578799*z^2 + 3042*z + 1
18
5:
171*z^14 + 322506*z^13 + 39857877*z^12 + 1178314830*z^11 + 12609530955*z^10 + 59047404052*z^9 + 133217516313*z^8 + 151416272538*z^7 + 87401792079*z^6 + 24964466832*z^5 + 3298273497*z^4 + 176587500*z^3 + 2953359*z^2 + 8550*z + 1
18
6:
z^15 + 80562*z^14 + 31892187*z^13 + 1958604477*z^12 + 36965679885*z^11 + 282834936723*z^10 + 1006775048306*z^9 + 1789315745103*z^8 + 1635346024965*z^7 + 766515158372*z^6 + 177851049531*z^5 + 18894963615*z^4 + 792932361*z^3 + 9837747*z^2 + 18546*z + 1
18
7:
816*z^15 + 3092742*z^14 + 578742714*z^13 + 23923991301*z^12 + 346131204768*z^11 + 2163041362350*z^10 + 6514394978760*z^9 + 10000101157164*z^8 + 7986505934988*z^7 + 3287595156969*z^6 + 669254167962*z^5 + 61882317576*z^4 + 2218218174*z^3 + 22590297*z^2 + 31806*z + 1
18
8:
18564*z^15 + 22828824*z^14 + 2927157291*z^13 + 98141557566*z^12 + 1230863521572*z^11 + 6892852400928*z^10 + 18951970865524*z^9 + 26850334989897*z^8 + 19910470050252*z^7 + 7628187583152*z^6 + 1443856347423*z^5 + 123509398104*z^4 + 4049593596*z^3 + 36828936*z^2 + 43740*z + 1
18
9:
z^16 + 48602*z^15 + 43277802*z^14 + 4937642026*z^13 + 155012076974*z^12 + 1858232498322*z^11 + 10049989911874*z^10 + 26842765525442*z^9 + 37068418696464*z^8 + 26842765525442*z^7 + 10049989911874*z^6 + 1858232498322*z^5 + 155012076974*z^4 + 4937642026*z^3 + 43277802*z^2 + 48602*z + 1
19
2:
19*z^9 + 969*z^8 + 11628*z^7 + 50388*z^6 + 92378*z^5 + 75582*z^4 + 27132*z^3 + 3876*z^2 + 152*z + 1
19
3:
190*z^12 + 30039*z^11 + 955415*z^10 + 10480685*z^9 + 49293030*z^8 + 110282061*z^7 + 122755200*z^6 + 68198866*z^5 + 18217200*z^4 + 2135030*z^3 + 91257*z^2 + 950*z + 1
19
4:
19*z^14 + 33269*z^13 + 4025834*z^12 + 120858525*z^11 + 1342914034*z^10 + 6639152463*z^9 + 16048390983*z^8 + 19836286558*z^7 + 12661568004*z^6 + 4082529468*z^5 + 626099647*z^4 + 40568477*z^3 + 881961*z^2 + 3857*z + 1
19
5:
19*z^15 + 130625*z^14 + 29557730*z^13 + 1369432315*z^12 + 21675013401*z^11 + 147061657389*z^10 + 480913930576*z^9 + 805588732316*z^8 + 708537813815*z^7 + 325700719261*z^6 + 75565120134*z^5 + 8211491763*z^4 + 363485143*z^3 + 5001503*z^2 + 11609*z + 1
19
6:
30039*z^15 + 25120679*z^14 + 2493211654*z^13 + 69598483653*z^12 + 760551811357*z^11 + 3829324392311*z^10 + 9693055120580*z^9 + 12889575031720*z^8 + 9126387532005*z^7 + 3394567246969*z^6 + 635015797444*z^5 + 54859737951*z^4 + 1872545361*z^3 + 18659235*z^2 + 27113*z + 1
19
7:
171*z^16 + 2352713*z^15 + 775920784*z^14 + 48255918534*z^13 + 990731824636*z^12 + 8594871485071*z^11 + 35842709418778*z^10 + 77065773283965*z^9 + 88367396632750*z^8 + 54385072015327*z^7 + 17621942820839*z^6 + 2861248364819*z^5 + 212171856865*z^4 + 6073952205*z^3 + 48409473*z^2 + 50369*z + 1
19
8:
11628*z^16 + 31878162*z^15 + 6395409234*z^14 + 304097767654*z^13 + 5211792552439*z^12 + 39420647699488*z^11 + 146881764495810*z^10 + 286321115707768*z^9 + 300232660601038*z^8 + 169724227366879*z^7 + 50559857331148*z^6 + 7526441514554*z^5 + 507820784433*z^4 + 13024616793*z^3 + 90102636*z^2 + 75563*z + 1
19
9:
75582*z^16 + 109361511*z^15 + 17545780527*z^14 + 737071348587*z^13 + 11612484687381*z^12 + 82368474208539*z^11 + 291074747871819*z^10 + 541769568970868*z^9 + 544581671136148*z^8 + 295702192439491*z^7 + 84635745299443*z^6 + 12087500823853*z^5 + 779463518167*z^4 + 18959730127*z^3 + 122436247*z^2 + 92359*z + 1
20
2:
z^10 + 190*z^9 + 4845*z^8 + 38760*z^7 + 125970*z^6 + 184756*z^5 + 125970*z^4 + 38760*z^3 + 4845*z^2 + 170*z + 1
20
3:
20*z^13 + 8455*z^12 + 484500*z^11 + 8533260*z^10 + 61734800*z^9 + 210549165*z^8 + 362435280*z^7 + 323168150*z^6 + 148060616*z^5 + 33330370*z^4 + 3346280*z^3 + 123880*z^2 + 1120*z + 1
20
4:
z^15 + 8835*z^14 + 2038965*z^13 + 97634255*z^12 + 1621045135*z^11 + 11688379531*z^10 + 41123373615*z^9 + 75065973395*z^8 + 72980758715*z^7 + 37724560705*z^6 + 10060714131*z^5 + 1295364045*z^4 + 71121845*z^3 + 1313185*z^2 + 4825*z + 1
20
5:
z^16 + 42084*z^15 + 18338230*z^14 + 1342518340*z^13 + 31243228095*z^12 + 302786852432*z^11 + 1404380739458*z^10 + 3363552566740*z^9 + 4322264895445*z^8 + 3012555338660*z^7 + 1123092670108*z^6 + 214763902832*z^5 + 19431933480*z^4 + 718919340*z^3 + 8223770*z^2 + 15484*z + 1
20
6:
8455*z^16 + 16358810*z^15 + 2672057400*z^14 + 110370179600*z^13 + 1708572381210*z^12 + 11990710860130*z^11 + 42310204249200*z^10 + 79444217584300*z^9 + 81480909758520*z^8 + 45737438804910*z^7 + 13749669039640*z^6 + 2105483766976*z^5 + 149867640070*z^4 + 4211021110*z^3 + 34105760*z^2 + 38740*z + 1
20
7:
20*z^17 + 1409495*z^16 + 858453820*z^15 + 81311662250*z^14 + 2369117497760*z^13 + 28327874751315*z^12 + 161408797284160*z^11 + 476419577424490*z^10 + 762141773678000*z^9 + 673179093472485*z^8 + 327040778272140*z^7 + 85063329007060*z^6 + 11200773586700*z^5 + 675949071650*z^4 + 15667137060*z^3 + 99090130*z^2 + 77500*z + 1
20
8:
4845*z^17 + 34783870*z^16 + 11352891781*z^15 + 775850189385*z^14 + 18211798035045*z^13 + 185148100760965*z^12 + 924538464224375*z^11 + 2435692164025601*z^10 + 3517044485217165*z^9 + 2822213885064205*z^8 + 1249029332944455*z^7 + 295743795079555*z^6 + 35286168261971*z^5 + 1910053052095*z^4 + 38942008125*z^3 + 208340035*z^2 + 125950*z + 1
20
9:
77520*z^17 + 209308845*z^16 + 49305029640*z^15 + 2821775678600*z^14 + 58799798443360*z^13 + 546323521110355*z^12 + 2534708717545540*z^11 + 6266670109931296*z^10 + 8543959393047340*z^9 + 6496023454577335*z^8 + 2727784026259000*z^7 + 612462511673170*z^6 + 69095814064036*z^5 + 3515037613920*z^4 + 66569397120*z^3 + 323168340*z^2 + 167940*z + 1
20
10:
z^18 + 184736*z^17 + 373684439*z^16 + 79469260064*z^15 + 4298493363980*z^14 + 86221454090612*z^13 + 778285461926792*z^12 + 3526521217218608*z^11 + 8542144050644948*z^10 + 11432612114864640*z^9 + 8542144050644948*z^8 + 3526521217218608*z^7 + 778285461926792*z^6 + 86221454090612*z^5 + 4298493363980*z^4 + 79469260064*z^3 + 373684439*z^2 + 184736*z + 1
Definition
The hypersimplex $\Delta(k,n)=\{x\in[0,1]^n:x_1+\cdots+x_n=k\}$ is a lattice polytope [2]. The table stores its Ehrhart $h^*$-polynomial $h^*_{\Delta(k,n)}(z)$, the numerator of its Ehrhart series [3].
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)
$\sum_{t\geq0}L_{\Delta(k,n)}(t)z^t= h^*_{\Delta(k,n)}(z)/(1-z)^n$.
(2)
$L_{\Delta(k,n)}(t)$ is the coefficient of $q^{kt}$ in $(1+q+\cdots+q^t)^n$.
(3)
$h^*_{\Delta(k,n)}(1)$ 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
(4)
The polynomial $h^*_{\Delta(k,n)}(z)$ is also called the Ehrhart $\delta$-polynomial. Formula (1) relates it to the Ehrhart polynomial $L_{\Delta(k,n)}(t)$.
(5)
The hypersimplex $\Delta(k,n)$ is the matroid polytope of the uniform matroid $U_{k,n}$ [2] and the moment polytope for the torus action on the Grassmannian $\mathrm{Gr}(k,n)$ [1].
(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 (3) 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 $h^*(z)=1$.
Programs
(P1)
Sage
n, k = 6, 2
T = PolynomialRing(QQ, "t"); t = T.gen()
Z = PolynomialRing(QQ, "z"); z = Z.gen()

L = sum((-1)**i * binomial(n, i) * binomial((k - i)*t - i + n - 1, n - 1)
        for i in range(k))
h = (sum(L(j)*z**j for j in range(n)) * (1 - z)**n).truncate(n)
h
References
[1]
Alexander Postnikov, Positive Grassmannian and polyhedral subdivisions, 2018. (arXiv)
Links
Similar tables
Ehrhart polynomials of the hypersimplices —   gives the counting polynomial $L_{\Delta(k,n)}(t)$ whose Ehrhart series has this numerator; the two tables determine each other by (1). A reader holding this numerator wants this table; a reader holding the Ehrhart counting polynomial wants T234.
Ehrhart $h^*$-polynomials of the Birkhoff polytopes —   the transportation-polytope $h^*$-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 $h^*_{\Delta(k,n)}(z)$ for every pair with $4\leq n\leq20$ and $2\leq k\leq\lfloor n/2\rfloor$, matching the companion Ehrhart polynomial table)
How they were obtained:

The $h^*$-polynomials are obtained exactly from the generating-function relation (1).

more

The rows were checked by direct coefficient counts from (2); their normalized volumes were checked against Eulerian numbers through (3); the symmetry $\Delta(k,n)\cong\Delta(n-k,n)$ was checked on the computed polynomials; and the rows with $k=2$ were compared with OEIS A275514 [4], whose title calls them Ehrhart coefficients but whose rows are the $h^*$-vectors.



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