OFFSET
1,4
COMMENTS
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
REFERENCES
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.
LINKS
Paul Barry, On the f-Matrices of Pascal-like Triangles Defined by Riordan Arrays, arXiv:1805.02274 [math.CO], 2018.
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.
L. Carlitz and Richard Scoville, Generalized Eulerian numbers: combinatorial applications, Journal für die reine und angewandte Mathematik 265 (1974), 111.
Colin Defant, Troupes, Cumulants, and Stack-Sorting, arXiv:2004.11367 [math.CO], 2020.
D. Dominici, Nested derivatives: A simple method for computing series expansions of inverse functions, arXiv:math/0501052 [math.CA], 2005.
D. Foata and M. P. Schützenberger, Théorie géometrique des polynômes eulériens, arXiv:math/0508232 [math.CO], 2005; Lecture Notes in Math. 138 (1970), 81-83.
Shi-Mei Ma, On gamma-vectors and the derivatives of the tangent and secant functions, arXiv:1304.6654 [math.CO], 2013.
Shi-Mei Ma, Jun Ma, and Yeong-Nan Yeh, On certain combinatorial expansions of descent polynomials and the change of grammars, arXiv:1802.02861 [math.CO], 2018.
Shi-Mei Ma and Yeong-Nan Yeh, The alternating run polynomials of permutations, arXiv:1904.11437 [math.CO], 2019. See p. 4.
A. Postnikov, V. Reiner, and L. Williams, Faces of generalized permutohedra, arXiv:math/0609184 [math.CO], 2006-2007. [Peter Bala, Oct 28 2008]
L. W. Shapiro, W.-J. Woan and S. Getu, Runs, slides and moments, SIAM J. Alg. Discrete Methods, 4 (1983), 459-466.
Andrei K. Svinin, Somos-4 equation and related equations, arXiv:2307.05866 [math.CA], 2023. See p. 16.
FORMULA
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).
From Peter Bala, Jun 26 2012: (Start)
T(n,k) = 2^k*A094503(n,k+1).
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.
Row sums A080635.
(End)
EXAMPLE
Triangle begins:
1;
1,
1, 2;
1, 8,
1, 22, 16;
1, 52, 136,
1, 114, 720, 272;
...
From Peter Bala, Jun 2012: (Start)
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)
MAPLE
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
MATHEMATICA
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 *)
CROSSREFS
KEYWORD
nonn,tabf,easy
AUTHOR
Don Knuth, Jan 28 2005
EXTENSIONS
More terms from Emeric Deutsch, Aug 06 2005
STATUS
approved