{ "StatisticIdentifier" : 156, "StatisticCollection" : "Permutations", "StatisticDescription" : "The Denert index of a permutation.\r\n\r\nIt is defined as\r\n$$\r\n\\begin{align*}\r\n den(\\sigma) &= \\#\\{ 1\\leq l < k \\leq n : \\sigma(k) < \\sigma(l) \\leq k \\} \\\\\r\n &+ \\#\\{ 1\\leq l < k \\leq n : \\sigma(l) \\leq k < \\sigma(k) \\} \\\\\r\n &+ \\#\\{ 1\\leq l < k \\leq n : k < \\sigma(k) < \\sigma(l) \\}\r\n\\end{align*}\r\n$$\r\nwhere $n$ is the size of $\\sigma$. It was studied by Denert in [1], and it was shown by Foata and Zeilberger in [2] that the bistatistic $(exc,den)$ is [[Permutations/Descents-Major#Euler-Mahonian_statistics|Euler-Mahonian]]. Here, $exc$ is the number of weak exceedences, see [[St000155]].", "StatisticTitle" : "The Denert index of a permutation.", "StatisticReferences" : "[1] Denert, M. The genus zeta function of hereditary orders in central simple algebras over global fields [[MathSciNet:0993928]]\r\n[2] Foata, D., Zeilberger, D. Denert's permutation statistic is indeed Euler-Mahonian [[MathSciNet:1061147]]\r\n[3] Triangle of Mahonian numbers T(n,k): coefficients in expansion of Product_i=0..n-1 (1 + x + ... + x^i), where k ranges from 0 to A000217(n-1). [[OEIS:A008302]]", "StatisticCode" : "import itertools\r\ndef statistic(pi):\r\n return sum(1 for l, k in itertools.combinations(range(pi.size()), 2)\r\n if (pi[k] < pi[l] <= k+1)\r\n or (pi[l] <= k+1 < pi[k])\r\n or (k+1 < pi[k] < pi[l]))", "StatisticSageCode" : "import itertools\r\ndef statistic(pi):\r\n return sum(1 for l, k in itertools.combinations(range(pi.size()), 2)\r\n if (pi[k] < pi[l] <= k+1)\r\n or (pi[l] <= k+1 < pi[k])\r\n or (k+1 < pi[k] < pi[l]))", "StatisticOriginalAuthor" : {"Name": "Christian Stump", "Time": "2013-07-24--12:25:21"}, "StatisticUpdateAuthor" : {"Name": "Martin Rubey", "Time": "2023-02-27--12:37:01"}, "StatisticGeneratingFunction" : { "1": { "0": 1 }, "2": { "0": 1, "1": 1 }, "3": { "0": 1, "1": 2, "2": 2, "3": 1 }, "4": { "0": 1, "1": 3, "2": 5, "3": 6, "4": 5, "5": 3, "6": 1 }, "5": { "0": 1, "1": 4, "2": 9, "3": 15, "4": 20, "5": 22, "6": 20, "7": 15, "8": 9, "9": 4, "10": 1 }, "6": { "0": 1, "1": 5, "2": 14, "3": 29, "4": 49, "5": 71, "6": 90, "7": 101, "8": 101, "9": 90, "10": 71, "11": 49, "12": 29, "13": 14, "14": 5, "15": 1 } }, "StatisticData" : { "[1]": 0, "[1,2]": 0, "[2,1]": 1, "[1,2,3]": 0, "[1,3,2]": 2, "[2,1,3]": 1, "[2,3,1]": 3, "[3,1,2]": 1, "[3,2,1]": 2, "[1,2,3,4]": 0, "[1,2,4,3]": 3, "[1,3,2,4]": 2, "[1,3,4,2]": 5, "[1,4,2,3]": 2, "[1,4,3,2]": 3, "[2,1,3,4]": 1, "[2,1,4,3]": 4, "[2,3,1,4]": 3, "[2,3,4,1]": 6, "[2,4,1,3]": 3, "[2,4,3,1]": 4, "[3,1,2,4]": 1, "[3,1,4,2]": 4, "[3,2,1,4]": 2, "[3,2,4,1]": 5, "[3,4,1,2]": 3, "[3,4,2,1]": 4, "[4,1,2,3]": 1, "[4,1,3,2]": 2, "[4,2,1,3]": 2, "[4,2,3,1]": 3, "[4,3,1,2]": 4, "[4,3,2,1]": 5, "[1,2,3,4,5]": 0, "[1,2,3,5,4]": 4, "[1,2,4,3,5]": 3, "[1,2,4,5,3]": 7, "[1,2,5,3,4]": 3, "[1,2,5,4,3]": 4, "[1,3,2,4,5]": 2, "[1,3,2,5,4]": 6, "[1,3,4,2,5]": 5, "[1,3,4,5,2]": 9, "[1,3,5,2,4]": 5, "[1,3,5,4,2]": 6, "[1,4,2,3,5]": 2, "[1,4,2,5,3]": 6, "[1,4,3,2,5]": 3, "[1,4,3,5,2]": 7, "[1,4,5,2,3]": 5, "[1,4,5,3,2]": 6, "[1,5,2,3,4]": 2, "[1,5,2,4,3]": 3, "[1,5,3,2,4]": 3, "[1,5,3,4,2]": 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