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A101280
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Triangle T(n,k) (n >= 1, 0 <= k <= floor((n-1)/2)) read by rows, where T(n,k) = (k+1)T(n-1,k) + (2n-4k)T(n-1,k-1).
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4
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1, 1, 1, 2, 1, 8, 1, 22, 16, 1, 52, 136, 1, 114, 720, 272, 1, 240, 3072, 3968, 1, 494, 11616, 34304, 7936, 1, 1004, 40776, 230144, 176896, 1, 2026, 136384, 1328336, 2265344, 353792, 1, 4072, 441568, 6949952, 21953408, 11184128, 1, 8166, 1398000, 33981760
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OFFSET
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1,4
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COMMENTS
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Row n has ceiling(n/2) terms.
The paper by Shapiro et al. explains why T(n,k) is the number of permutations of n elements having k peaks and with the further property that every rise (ascent) is immediately followed by a peak. [That is, the permutation p_1 ... p_n has the further property that (j > 1 and p_{j-1} < p_j) implies (j < n and p_j > p_{j+1}).] For example, the T(4,1)=8 permutations in the case n=4, k=1 are 1423, 2143, 2431, 3142, 3241, 3421, 4231, 4132.
A more elegant way to state this property: T(n,k) is the number of permutations of n objects with k descents such that every descent is a peak. The eight examples for n=4 and k=1 are now 1243, 1324, 1342, 1423, 2314, 2341, 2413, 3412.
The rows of this triangle are the gamma vectors of the n-dimensional (type A) permutohedra (Postnikov et al., p. 31). Cf. A055151 and A089627. - Peter Bala, Oct 28 2008
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REFERENCES
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D. Foata and V. Strehl, "Euler numbers and variations of permutations", in Colloquio Internazionale sulle Teorie Combinatorie, Rome, September 1973, (Atti dei Convegni Lincei 17, Rome, 1976), 129.
Guoniu Han, Frédéric Jouhet, Jiang Zeng, Two new triangles of q-integers via q-Eulerian polynomials of type A and B, Ramanujan J (2013) 31:115-127, DOI 10.1007/s11139-012-9389-3
T. K. Petersen, Eulerian Numbers, Birkhauser, 2015, Chapter 4.
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LINKS
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F. Bergeron, Ph. Flajolet and B. Salvy, Varieties of Increasing Trees, Lecture Notes in Computer Science vol. 581, ed. J.-C. Raoult, Springer 1992, pp. 24-48.
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FORMULA
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G.f.: Sum_{n>=1, k>=0} T(n, k) t^k z^n/n! = C(t)(2-C(t))/(exp^(-z sqrt(1-4t)) + 1 - C(t)) - C(t), where the sum on k is actually finite, running up to ceiling(n/2) - 1; here C(t) = (1 - sqrt(1-4t))/2t is the generating function for the Catalan numbers (A000108).
Sum_{k} Eulerian(n, k) x^k = Sum_{k} T(n, k) x^k (1+x)^(n-1-2k). E.g., 1 + 11x + 11x^2 + x^3 = (1+x)^3 + 8x(1+x).
Let r(t) = sqrt(1 - 2*t) and w(t) = (1 - r(t))/(1 + r(t)). Define F(t,z) = r(t)*(1 + w(t)*exp(r(t)*z))/(1 - w(t)*exp(r(t)*z)) = 1 + t*z + t*z^2/2! + (t+t^2)*z^3/3! + (t+4*t^2)*z^4/4! + ...; F(t,z) is the e.g.f. for A094503. The e.g.f. for the present table is A(t,z) := (F(2*t,z) - 1)/(2*t) = z + z^2/2! + (1+2*t)*z^3/3! + (1+8*t)*z^4/4! + ....
The e.g.f. A(t,z) satisfies the autonomous partial differential equation dA/dz = 1 + A + t*A^2 with A(t,0) = 0. It follows that the inverse function A(t,z)^(-1) may be expressed as an integral: A(t,z)^(-1) = int {x = 0..z} 1/(1+x+t*x^2) dx.
Applying [Dominici, Theorem 4.1] to invert the integral gives the following method for calculating the row polynomials R(n,t) of the table: let f(t,x) = 1+x+t*x^2 and let D be the operator f(t,x)*d/dx. Then R(n+1,t) = D^n(f(t,x)) evaluated at x = 0.
By Bergeron et al., Theorem 1, the row polynomial R(n,t) is the generating function for rooted plane increasing 0-1-2 trees on n vertices, where the vertices of outdegree 2 have weight t and all other vertices have weight 1. An example is given below.
(End)
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EXAMPLE
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Triangle begins:
1;
1,
1, 2;
1, 8,
1, 22, 16;
1, 52, 136,
1, 114, 720, 272;
...
n = 4: the 9 weighted plane increasing 0-1-2 trees on 4 vertices are
..................................................................
..4...............................................................
..|...............................................................
..3..........4...4...............4...4...............3...3........
..|........./.....\............./.....\............./.....\.......
..2....2...3.......3...2...3...2.......2...3...4...2.......2...4..
..|.....\./.........\./.....\./.........\./.....\./.........\./...
..1...(t)1........(t)1....(t)1........(t)1....(t)1........(t)1....
..................................................................
..3...4...4...3...................................................
...\./.....\./....................................................
.(t)2....(t)2.....................................................
....|.......|.....................................................
....1.......1.....................................................
Hence R(4,t) = 1 + 8*t.
(End)
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MAPLE
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T:=proc(n, k) if k<0 then 0 elif n=1 and k=0 then 1 elif k>floor((n-1)/2) then 0 else (k+1)*T(n-1, k)+(2*n-4*k)*T(n-1, k-1) fi end: for n from 1 to 13 do seq(T(n, k), k=0..floor((n-1)/2)) od; # yields sequence in triangular form # Emeric Deutsch, Aug 06 2005
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MATHEMATICA
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t[_, k_?Negative] = 0; t[1, 0] = 1; t[n_, k_] /; k > (n-1)/2 = 0; t[n_, k_] := t[n, k] = (k+1)*t[n-1, k] + (2*n-4*k)*t[n-1, k-1]; Table[t[n, k], {n, 1, 13}, {k, 0, (n-1)/2}] // Flatten (* Jean-François Alcover, Nov 22 2012 *)
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CROSSREFS
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The numbers 2^{n-1-k} T(n, k) form the array shown in A008303.
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KEYWORD
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nonn,tabf,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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