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%I #90 Nov 06 2023 13:16:48
%S 1,1,2,1,6,6,1,8,6,36,24,1,10,20,60,90,240,120,1,12,30,20,90,360,90,
%T 480,1080,1800,720,1,14,42,70,126,630,420,630,840,5040,2520,4200,
%U 12600,15120,5040,1,16,56,112,70,168,1008,1680,1260,1680,1344,10080,6720
%N Irregular triangle read by rows: Row n gives numbers of preferential arrangements (onto functions) of n objects that are associated with the partition of n, taken in Abramowitz and Stegun order.
%C This is a refinement of A019538 with row sums in A000670.
%C From _Tom Copeland_, Sep 29 2008: (Start)
%C This array is related to the reciprocal of an e.g.f. as sketched in A133314. For example, the coefficient of the fourth-order term in the Taylor series expansion of 1/(a(0) + a(1) x + a(2) x^2/2! + a(3) x^3/3! + ...) is a(0)^(-5) * {24 a(1)^4 - 36 a(1)^2 a(2) a(0) + [8 a(1) a(3) + 6 a(2)^2] a(0)^2 - a(4) a(0)^3}.
%C The unsigned coefficients characterize the P3 permutohedron depicted on page 10 in the Loday link with 24 vertices (0-D faces), 36 edges (1-D faces), 6 squares (2-D faces), 8 hexagons (2-D faces) and 1 3-D permutohedron. Summing coefficients over like dimensions gives A019538 and A090582. Compare to A133437 for the associahedron.
%C Given the n X n lower triangular matrix M = [ binomial(j,k) u(j-k) ], the first column of the inverse matrix M^(-1) contains the (n-1) rows of A049019 as the coefficients of the multinomials formed from the u(j). M^(-1) can be computed as (1/u(0)){I - [I- M/u(0)]^n} / {I - [I- M/u(0)]} = - u(0)^(-n) {sum(j=1 to n)(-1)^j bin(n,j) u(0)^(n-j) M^(j-1)} where I is the identity matrix.
%C Another method for computing the coefficients and partitions up to (n-1) rows is to use (1-x^n)/ (1-x) = 1+x^2+x^3+ ... + x^(n-1) with x replaced either by [I- M/a(0)] or [1- g(x)/a(0)] with the n X n matrix M = [bin(j,k) a(j-k)] and g(x)= a(0) + a(1)x + a(2)x^2/2! + ... + a(n) x^n/n!. The first n terms (rows of the first column) of the resulting series (matrix) divided by a(0) contain the (n-1) rows of signed coefficients and associated partitions for A049019.
%C To obtain unsigned coefficients, change a(j) to -a(j) for j>0. A133314 contains other matrices and recursion formulas that could be used. The Faa di Bruno formula gives the coefficients as n! [e(1)+e(2)+...+e(n)]! / [1!^e(1) e(1)! 2!^e(2) e(2)!... n!^e(n) e(n)! ] for the partition of form [a(1)^e(1)...a(n)^e(n)] with [e(1)+2e(2)+...+ n e(n)] = n (see Abramowitz and Stegun pages 823 and 831) in agreement with Arnold's formula. (End)
%H Peter Luschny, <a href="/A049019/b049019.txt">Rows n = 1..36, flattened</a>
%H M. Abramowitz and I. A. Stegun, eds., <a href="https://personal.math.ubc.ca/~cbm/aands/">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972.
%H N. Arkani-Hamed, Y. Bai, S. He, and G. Yan, <a href="https://arxiv.org/abs/1711.09102">Scattering forms and the positive geometry of kinematics, color, and the worldsheet </a>, arXiv:1711.09102 [hep-th], 2017.
%H Tom Copeland, <a href="http://math.stackexchange.com/questions/1785526/bijective-mapping-between-face-polytopes-of-permutohedra-and-partitions-of-integ">Bijective mapping between face polytopes of permutohedra and partitions of integers</a>, Math StackExchange question, 2016.
%H S. Forcey, <a href="https://sforcey.github.io/sf34/hedra.htm">The Hedra Zoo</a>.
%H X. Gao, S. He, and Y. Zhan, <a href="https://arxiv.org/abs/1708.08701">Labelled tree graphs, Feynman diagrams and disk integrals </a>, arXiv:1708.08701 [hep-th], 2017.
%H J. Loday, <a href="https://web.archive.org/web/20100202074906/http://www-irma.u-strasbg.fr/~loday/PAPERS/MultFAsENG2.pdf">The Multiple Facets of the Associahedron</a>
%H V. Pilaud, <a href="https://www.mat.univie.ac.at/~slc/wpapers/s76vortrag/pilaud.pdf">The Associahedron and its Friends</a>, presentation for Seminaire Lotharingien de Combinatoire, April 4 - 6, 2016.
%H A. Postnikov, <a href="https://arxiv.org/abs/1806.05307">Positive Grassmannian and Polyhedral Subdivisions</a>, arXiv:1806.05307 [math.CO], (cf. p. 17), 2018.
%F a(n) = A048996(n) * A036038(n);
%F a(n) = A036040(n) * factorial(A036043(n)).
%F A lowering operator for the unsigned multinomials in the brackets in the example is [d/du(1) 1/POP] where u(1) is treated as a continuous variable and POP is an operator that pulls off the number of parts of a partition ignoring u(0), e.g., [d/du(1) 1/POP][ u(0)u(2) + 2 u(1)^2 ] = (1/2) 2*2 u(1) = 2*u(1), analogous to the prototypical delta operator (d/dz) z^n = n z^(n-1). - _Tom Copeland_, Oct 04 2008
%F From the matrix formulation with M_m,k = 1/(m-k)!; g(x) = exp[ u(.) x]; an orthonormal vector basis x_1, ..., x_n and En(x^k) = x_k for k <= n and zero otherwise, for j=0 to n-1 the j-th signed row multinomial is given by the wedge product of x_1 with the wedge product (-1)^j * j! * u(0)^(-n) * Wedge{ En[x g(x), x^2 g(x), ..., x^(j) g(x), ~, x^(j+2) g(x), ..., x^n g(x)] } where Wedge{a,b,c} = a v b v c (the usual wedge symbol is inverted here to prevent confusion with the power notation, see Mathworld) and the (j+1)-th element is omitted from the product. _Tom Copeland_, Oct 06 2008 [Changed an x^n to x^(n-1) and "inner product of x_1" to "wedge". - _Tom Copeland_, Feb 03 2010]
%e Irregular triangle starts (note the grouping by ';' when comparing with A019538):
%e [1] 1;
%e [2] 1; 2;
%e [3] 1; 6; 6;
%e [4] 1; 8, 6; 36; 24;
%e [5] 1; 10, 20; 60, 90; 240; 120;
%e [6] 1; 12, 30, 20; 90, 360, 90; 480, 1080; 1800; 720;
%e [7] 1; 14, 42, 70; 126, 630, 420, 630; 840, 5040, 2520; 4200, 12600; 15120; 5040;
%e .
%e a(17) = 240 because we can write
%e A048996(17)*A036038(17) = 4*60 = A036040(17)*A036043(17)! = 10*24.
%e As in A133314, 1/exp[u(.)*x] = u(0)^(-1) [ 1 ] + u(0)^(-2) [ -u(1) ] x + u(0)^(-3) [ -u(0)u(2) + 2 u(1)^2 ] x^2/2! + u(0)^(-4) [ -u(0)^2 u(3) + 6 u(0)u(1)u(2) - 6 u(1)^3 ] x^3/3! + u(0)^(-5) [ -u(0)^3 u(4) + 8 u(0)^2 u(1)u(3) + 6 u(0)^2 u(2)^2 - 36 u(0)u(1)^2 u(2) + 24 u(1)^4 ] x^4/4! + ... . These are essentially refined face polynomials for permutohedra: empty set + point + line segment + hexagon + 3-D- permutohedron + ... . - _Tom Copeland_, Oct 04 2008
%o (SageMath)
%o def A049019(n):
%o if n == 0: return [1]
%o P = lambda k: Partitions(n, min_length=k, max_length=k)
%o Q = (p.to_list() for k in (1..n) for p in P(k))
%o return [factorial(len(p))*SetPartitions(sum(p), p).cardinality() for p in Q]
%o for n in (1..7): print(A049019(n)) # _Peter Luschny_, Aug 30 2019
%Y Cf. A000041, A000670, A019538, A036038, A36040, A036043, A048996, A133314.
%K nonn,tabf
%O 1,3
%A _Alford Arnold_
%E Partitions for 7 and 8 from _Tom Copeland_, Oct 02 2008
%E Definition edited by _N. J. A. Sloane_, Nov 06 2023
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