The sequence A000140 is studied
 <http://oeis.org/A000140>
 (Kendall-Mann numbers: the maximum number of permutations on n letters having the same number of inversions )
 and I am looking for a proof that
 M(n)/M(n-1)=n+1/2 when n= infinity, M(n) - max element in row n.
 If you have any ideas how to proove or disproove it, even the question is too hard, could you let me know in anyway.
 Modelling with Pari GP shows
 for n = 0 to 149 M(n)/M(n-1):
 1.00000000, 1.00000000, 2.00000000, 3.00000000, 3.66666667, 4.59090909, 5.67326733, 6.69458988, 7.61939520, 8.57906801, 9.60953383, 10.6235009, 11.5884536, 12.5657349, 13.5817521, 14.5907723, 15.5704306, 16.5558579, 17.5656455, 18.5718445, 19.5585507, 20.5484134, 21.5549876, 22.5594838, 23.5501133, 24.5426559, 25.5473665, 26.5507683, 27.5438066, 28.5380914, 29.5416285, 30.5442887, 31.5389122, 32.5343930, 33.5371446, 34.5392804, 35.5350028, 36.5313400, 37.5335406, 38.5352923, 39.5318079, 40.5287792, 41.5305788, 42.5320411, 43.5291478, 44.5266018, 45.5281005, 46.5293394, 47.5268986, 48.5247283, 49.5259956, 50.5270586, 51.5249718, 52.5230999, 53.5241854, 54.5251073, 55.5233026, 56.5216716, 57.5226117, 58.5234188, 59.5218427, 60.5204088, 61.5212309, 62.5219434, 63.5205550, 64.5192845, 65.5200094, 66.5206430, 67.5194106, 68.5182772, 69.5189212, 70.5194882, 71.5183871, 72.5173696, 73.5179455, 74.5184560, 75.5174660, 76.5165477, 77.5170657, 78.5175276, 79.5166329, 80.5157998, 81.5162682, 82.5166882, 83.5158756, 84.5151165, 85.5155421, 86.5159256, 87.5151843, 88.5144896, 89.5148781, 90.5152297, 91.5145507, 92.5139127, 93.5142686, 94.5145921, 95.5139679, 96.5133798, 97.5137071, 98.5140058, 99.5134299, 100.512886, 101.513188, 102.513465, 103.512932, 104.512428, 105.512707, 106.512964, 107.512470, 108.512001, 109.512260, 110.512499, 111.512039, 112.511602, 113.511843, 114.512067, 115.511637, 116.511229, 117.511454, 118.511663, 119.511261, 120.510879, 121.511090, 122.511285, 123.510909, 124.510550, 125.510748, 126.510932, 127.510578, 128.510241, 129.510426, 130.510599, 131.510267, 132.509949, 133.510123, 134.510286, 135.509973, 136.509673, 137.509838, 138.509992, 139.509696, 140.509412, 141.509568, 142.509713, 143.509434, 144.509165, 145.509312, 146.509450, 147.509185, 148.508930,


n+0.5 for n = infinity
gaichenkov@yandex.ru



The property of Kendall-Mann numbers 
It is known that 
 $$ \left| P\left( \frac{\mathrm{inv}(\pi)-\frac 12{n\choose 2}}{\sqrt{n(n-1)(2n+5)/72}}\leq x\right)-\Phi(x)\right| \leq \frac{C}{\sqrt{n}}, $$
where $\Phi(x)$ denotes the standard normal distribution. From this it is immediate that $M(n+1)/M(n)=n-\frac 12+o(1)$. 
For some references, see <http://arxiv.org/PS_cache/math/pdf/0508/0508242v2.pdf>.


