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A212806
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Number of n X n matrices in which each row is a permutation of [1..n] and which contain no column rises.
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5
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1, 3, 163, 271375, 21855093751, 128645361626874561, 78785944892341703819175577, 6795588328283070704898044776213094655, 107414633522643325764587104395687638119674465944431, 392471529081605251407320880492124164530148025908765037878553312273, 407934916447631403509359040563002566177814886353044858592046202746464825839911293037
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OFFSET
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1,2
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COMMENTS
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A column rise in a matrix M = (m_{i,j}) is a value of j such that m_{i,j} < m_{i,j+1} for all i = 1..n.
From Petros Hadjicostas, Aug 26 2019: (Start)
Let R(m,n) := R(m,n,t=0) = A212855(m,n) for m,n >= 1, where R(m,n,t) = LHS of Eq. (6) of Abramson and Promislow (1978, p. 248).
Let P_n be the set of all lists b = (b_1, b_2,..., b_n) of integers b_i >= 0, i = 1,..., n, such that 1*b_1 + 2*b_2 + ... + n*b_n = n; i.e., P_n is the set all integer partitions of n. Then |P_n| = A000041(n) for n >= 0.
We have a(n) = R(n,n) = A212855(n,n) = Sum_{b in P_n} (-1)^(n - Sum_{j=1..n} b_j) * (b_1 + b_2 + ... + b_n)!/(b_1! * b_2! * ... * b_n!) * (n! / ((1!)^b_1 * (2!)^b_2 * ... * (n!)^b_n)^n.
(End)
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LINKS
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Alois P. Heinz, Table of n, a(n) for n = 1..30 (first 18 terms from R. H. Hardin)
Milton Abramowitz and Irene A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, National Bureau of Standards (Applied Mathematics Series, 55), 1964; see pp. 831-832 for the multinomial coefficients of integer partitions of n = 1..10.
Morton Abramson, David Promislow, Enumeration of arrays by column rises, J. Combinatorial Theory Ser. A 24(2) (1978), 247-250. MR0469773 (57 #9554). [Their a(5) on p. 250 is wrong; see A212845.]
Wikipedia, Partition (number theory).
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FORMULA
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Abramson and Promislow give a g.f. for R(m,n,t), the number of m X n matrices in which each row is a permutation of [1..n] and which contain exactly t column rises:
1 + Sum_{n>=1} Sum_{t=0..n-1} R(m,n,t) y^t x^n/(n!)^m = (y-1)/(y-f(x(y-1))) where f(x) = Sum_{i>=0} x^i/(i!)^m.
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EXAMPLE
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For n=2 the three matrices are [12/21], [21/12], [21/21] (but not [12/12]).
From Petros Hadjicostas, Aug 26 2019: (Start)
For example, when n = 3, the integer partitions of 3 are 3, 1+2, and 1+1+1, with corresponding (b_1, b_2, b_3) notation (0,0,1), (1,1,0), and (3,0,0). The corresponding multinomial coefficients are 3!/3! = 1, 3!/(1!2!) = 3, and 3!/(1!1!1!) = 6, while the corresponding quantities (b_1 + b_2 + b_3)!/(b_1!*b_2!*b_3!) are 1, 2, and 1. The corresponding exponents of -1 (i.e., n - Sum_{j = 1..n} b_j}) are 3 -(0+0+1) = 2, 3 - (1+1+0) = 1, and 3 - (3+0+0) = 0.
It follows that a(n) = (-1)^2 * 1 * 1^3 + (-1)^1 * 2 * 3^3 + (-1)^0 * 1 * 6^3 = 163.
(End)
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MAPLE
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A212806 := proc(n) sum(z^k/k!^n, k=0..infinity);
series(%^x, z=0, n+1): n!^n*coeff(%, z, n); add(abs(coeff(%, x, k)), k=0..n) end:
seq(A212806(n), n=1..11); # Peter Luschny, May 27 2017
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MATHEMATICA
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a[n_] := Module[{s0, s1, s2}, s0 = Sum[z^k/k!^n, {k, 0, n}]; s1 = Series[s0^x, {z, 0, n + 1}] // Normal; s2 = n!^n*Coefficient[s1, z, n]; Sum[Abs[Coefficient[s2, x, k]], {k, 0, n}]]; Array[a, 11] (* Jean-François Alcover, Feb 27 2018, after Peter Luschny *)
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CROSSREFS
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A212805 is a lower bound.
Main diagonal of A212855.
Cf. A000041, A070289, A212850, A212851, A212852, A212853, A212854, A212856, A212857, A212858, A212859, A212860, A309951, A325305.
Sequence in context: A341224 A042439 A212845 * A030258 A154737 A285488
Adjacent sequences: A212803 A212804 A212805 * A212807 A212808 A212809
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KEYWORD
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nonn
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AUTHOR
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N. J. A. Sloane, May 27 2012
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EXTENSIONS
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Corrected by R. H. Hardin, May 28 2012
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STATUS
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approved
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