OFFSET
1,2
COMMENTS
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)
LINKS
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 and 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).
FORMULA
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.
EXAMPLE
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)
MAPLE
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
MATHEMATICA
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 *)
T[n_, k_] := T[n, k] = If[k == 0, 1, -Sum[Binomial[k, j]^n*(-1)^j*T[n, k-j], {j, 1, k}]];
a[n_] := T[n, n];
CROSSREFS
KEYWORD
nonn,changed
AUTHOR
N. J. A. Sloane, May 27 2012
EXTENSIONS
Corrected by R. H. Hardin, May 28 2012
STATUS
approved