Ehrhart polynomials
This page contains the Ehrhart polynomials E_P(t) of examples of flow polytopes.
CRY polytope: the flow polytope of the complete graph with n+1 vertices with netflow e_1 - e_(n+1)
the values up to n=6 were computed in [Sec 6, 1] and [2]. The values for n=7..13 were computed using the Maple code for "Volume computation for polytopes and partition functions for classical root systems" by Baldoni, Beck, Cochet, Vergne, link
Note that all these polynomials have positive coefficients.
Conjecture 1: The Ehrhart polynomial of the CRY polytope has positive coefficients.
n = 2
t+1
n = 3
(t+3)*(t+2)*(t+1)/3!
n = 4
(t+6)*(t+5)*(t+4)*(t+3)*(t+2)*(t+1)/6! +
(t+5)*(t+4)*(t+3)*(t+2)*(t+1)*t/6!
n = 5
(t+10)*(t+9)*(t+8)*(t+7)*(t+6)*(t+5)*(t+4)*(t+3)*(t+2)*(t+1)/10! +
5*(t+9)*(t+8)*(t+7)*(t+6)*(t+5)*(t+4)*(t+3)*(t+2)*(t+1)*t/10!+
4*(t+8)*(t+7)*(t+6)*(t+5)*(t+4)*(t+3)*(t+2)*(t+1)*t*(t-1)/10!
n = 6
(t+15)*(t+14)*(t+13)*(t+12)*(t+11)*(t+10)*(t+9)*(t+8)*(t+7)*(t+6)*(t+5)*(t+4)*(t+3)*(t+2)*(t+1)/15! +
16*(t+14)*(t+13)*(t+12)*(t+11)*(t+10)*(t+9)*(t+8)*(t+7)*(t+6)*(t+5)*(t+4)*(t+3)*(t+2)*(t+1)*t/15!+
58*(t+13)*(t+12)*(t+11)*(t+10)*(t+9)*(t+8)*(t+7)*(t+6)*(t+5)*(t+4)*(t+3)*(t+2)*(t+1)*t*(t-1)/15! +
56*(t+12)*(t+11)*(t+10)*(t+9)*(t+8)*(t+7)*(t+6)*(t+5)*(t+4)*(t+3)*(t+2)*(t+1)*t*(t-1)*(t-2)/15! +
9*(t+11)*(t+10)*(t+9)*(t+8)*(t+7)*(t+6)*(t+5)*(t+4)*(t+3)*(t+2)*(t+1)*t*(t-1)*(t-2)*(t-3)/15!
n = 7
(1/8688935743488000)*t^21+(2117/121645100408832000)*t^20+(149789/121645100408832000)*t^19+(593/10944228556800)*t^18+(1182547/711374856192000)*t^17+(2371609/62768369664000)*t^16+(124270847/188305108992000)*t^15+(26270857/2897001676800)*t^14+(468626303/4707627724800)*t^13+(8513133061/9656672256000)*t^12+(4077796979/643778150400)*t^11+(71471423831/1931334451200)*t^10+(66150911695291/376610217984000)*t^9+(7939938012827/11769069312000)*t^8+(97984316095277/47076277248000)*t^7+(615428916451/120708403200)*t^6+(433329666631051/44460928512000)*t^5+(26256060764993/1852538688000)*t^4+(88462713645601/5866372512000)*t^3+(41425488163/3760495200)*t^2+(571574671/116396280)*t+1
Data for n=9..13 available in the text file below "EhrhartCRYn=2-13.txt"
Tesler polytope: the flow polytope of the complete graph with n+1 vertices with netflow e_1+e_2+....+e_n-n*e_(n+1)
These values were computed using the Maple code for "Volume computation for polytopes and partition functions for classical root systems" by Baldoni, Beck, Cochet, Vergne, link
n = 2
t+1
n = 3
(2/3)*t^3+(5/2)*t^2+(17/6)*t+1
n = 4
(2/9)*t^6+(241/120)*t^5+(511/72)*t^4+(301/24)*t^3+(841/72)*t^2+(109/20)*t+1
n = 5
(4/135)*t^10+(31649/60480)*t^9+(160403/40320)*t^8+(21569/1260)*t^7+(44251/960)*t^6+(77783/960)*t^5+(1624423/17280)*t^4+(2147039/30240)*t^3+(22439/672)*t^2+(7421/840)*t+1
n = 6
(8/6075)*t^15+(249489293/6227020800)*t^14+(3428111017/6227020800)*t^13+(197140819/43545600)*t^12+(11922763249/479001600)*t^11+(602180419/6220800)*t^10+(11963740237/43545600)*t^9+(25199692817/43545600)*t^8+(7932207629/8709120)*t^7+(11644098569/10886400)*t^6+(5053221653/5443200)*t^5+(156138487/267300)*t^4+(2571500089/9979200)*t^3+(7390897/98280)*t^2+(1169603/90090)*t+1
n = 7
(32/1913625)*t^21+(43711098017/54305848396800)*t^20+(8289800133889/460776895488000)*t^19+(66208825046761/266765571072000)*t^18+(3241014459563/1368028569600)*t^17+(521821195268227/31384184832000)*t^16+(698207665056763/7846046208000)*t^15+(4375792097394217/11769069312000)*t^14+(9661087272979733/7846046208000)*t^13+(145909978681067/44706816000)*t^12+(933663678689257/134120448000)*t^11+(14452330182703183/1207084032000)*t^10+(1820069848644139/109734912000)*t^9+(1735099741442786993/94152554496000)*t^8+(63937653301265773/3923023104000)*t^7+(44483185033062973/3923023104000)*t^6+(76702307545121/12573792000)*t^5+(9168769987984879/3705077376000)*t^4+(2812490391119/3859455600)*t^3+(14316645224033/97772875200)*t^2+(297528821/16628040)*t+1
Data for n=8..10 available in the text file below "EhrhartTeslern=2-10.txt"
Note that all these polynomials have positive coefficients.
Conjecture 2: The Ehrhart polynomial of the Tesler polytope has positive coefficients.
Also the roots of the Ehrhart polynomial appear to have negative real parts.
Conjecture 3: The roots of the Ehrhart polynomial of the CRY and Tesler polytope have negative real parts.
Also the h* polynomial appears to have real roots.
Conjecture 4: The h* polynomials of the CRY and Tesler polytope have real roots.
References:
[1] De Loera, Jesus A., Fu Liu, and Ruriko Yoshida. "A generating function for all semi-magic squares and the volume of the Birkhoff polytope." Journal of Algebraic Combinatorics 30.1 (2009): 113-139.
[2] Moorefield, Dorothy L., Partition analysis in Ehrhart theory, Master thesis San Francisco State University (2007), link