OFFSET
1,3
COMMENTS
Number of entries in row n is ceiling(binomial(n,2)/2) - ceiling(binomial(n-2,2)/2).
Sum of entries in row n is A001710(n) for n>=2.
LINKS
Alois P. Heinz, Rows n = 1..143, flattened
J. Shareshian and M. L. Wachs, q-Eulerian polynomials: excedance number and major index, Electronic Research Announcements of the Amer. Math. Soc., 13 (2007), 33-45.
R. P. Stanley, Binomial posets, Möbius inversion and permutation enumeration, J. Combinat. Theory, A 20 (1976), 336-356.
S. Tanimoto, A study of Eulerian numbers for permutations in the alternating group, Integers, Electronic J. of Combinatorial Number Theory, 6 (2006), #A31.
FORMULA
In the Shareshian and Wachs reference (p. 35) a q-analog of the exponential g.f. of the Eulerian polynomials is given for the joint distribution of (inv, des) (see also the Stanley reference). The first Maple program given below makes use of this function by considering its odd part.
T(n,k) = (euler(n,k) - Sum_{j=max(0, k+1-ceiling(n/2))..min(floor(n/2), k)} binomial(j-1-floor(n/2), j) * euler(ceiling(n/2), k-j)) / 2, where euler(n,k) is the Eulerian number A173018 (not A008292, which has different indexing). - Robert A. Russell, Nov 16 2018
EXAMPLE
T(4,2) = 6 because we have 1432, 3142, 3214, 4312, 4231 and 3421.
Triangle begins with T(1,1):
0
1
2 1
6 6
12 36 12
28 155 147 29 1
56 605 1208 586 64 1
120 2160 7800 7800 2160 120
240 7320 44160 78000 44160 7320 240
496 23947 227623 655039 655315 227569 23893 517 1
992 76305 1102068 4868556 7862124 4869558 1101420 76332 1044 1
MAPLE
for n to 11 do qbr := proc (m) options operator, arrow; sum(q^i, i = 0 .. m-1) end proc; qfac := proc (m) options operator, arrow; product(qbr(j), j = 1 .. m) end proc; Exp := proc (z) options operator, arrow; sum(q^binomial(m, 2)*z^m/qfac(m), m = 0 .. 19) end proc; g := (1-t)/(Exp(z*(t-1))-t); gser := simplify(series(g, z = 0, 17)); a[n] := simplify(qfac(n)*coeff(gser, z, n)); b[n] := (a[n]-subs(q = -q, a[n]))*1/2; P[n] := sort(subs(q = 1, b[n])) end do; 0; for n to 11 do seq(coeff(P[n], t, j), j = 1 .. ceil((1/2)*binomial(n, 2))-ceil((1/2)*binomial(n-2, 2))) end do; # yields sequence in triangular form
# second Maple program:
b:= proc(u, o, t) option remember; `if`(u+o=0, t, expand(
add(b(u+j-1, o-j, irem(t+j-1+u, 2)), j=1..o)+
add(b(u-j, o+j-1, irem(t+u-j, 2))*x, j=1..u)))
end:
T:= n->`if`(n=1, 0, (p->seq(coeff(p, x, i), i=1..degree(p)))
(add(b(j-1, n-j, irem(j+1, 2)), j=1..n))):
seq(T(n), n=1..12); # Alois P. Heinz, Nov 19 2013
MATHEMATICA
b[u_, o_, t_] := b[u, o, t] = If[u+o == 0, t, Expand[Sum[b[u+j-1, o-j, Mod[t+j-1+u, 2]], {j, 1, o}] + Sum[b[u-j, o+j-1, Mod[t+u-j, 2]]*x, {j, 1, u}]]]; T[n_] := If[n == 1, 0, Function[{p}, Table[Coefficient[p, x, i], {i, 1, Exponent[p, x]}]][Sum[ b[j-1, n-j, Mod[j+1, 2]], {j, 1, n}]]]; Table[T[n], {n, 1, 12}] // Flatten (* Jean-François Alcover, May 28 2015, after Alois P. Heinz *)
Needs["Combinatorica`"];
Join[{0}, Table[(Eulerian[n, k] - Sum[Binomial[j-1-Floor[n/2], j] Eulerian[Ceiling[n/2], k-j], {j, Max[0, k+1-Ceiling[n/2]], Min[Floor[n/2], k]}])/2, {n, 2, 15}, {k, 1, n}] // Flatten // DeleteCases[0]] (* Robert A. Russell, Nov 16 2018 *)
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Emeric Deutsch, Nov 11 2008
STATUS
approved