A231074
The number of possible ways to arrange the sums x_i + x_j (1 <= i < j <= n) of the items x_1 < x_2 <...< x_n in nondecreasing order.
Original entry on oeis.org
1, 1, 1, 1, 2, 12, 244
Offset: 0
Let a < b < c < d. There are two possible ways to arrange the sums in nondecreasing order:
1) a+b <= a+c <= a+d <= b+c <= b+d <= c+d, (for instance, a = 1, b = 3, c = 4, d = 5);
2) a+b <= a+c <= b+c <= a+d <= b+d <= c+d, (for instance, a = 1, b = 2, c = 3, d = 5).
Hence a(4) = 2.
A231085
The number of possible ways to arrange the sums x_i + x_j (1 <= i < j <= n) of the items x_1 < x_2 <...< x_n in increasing order provided that all sums are different.
Original entry on oeis.org
1, 1, 1, 1, 2, 12, 168, 4680
Offset: 0
Let a < b < c < d. There are two possible ways to arrange the sums in increasing order:
1) a+b < a+c < a+d < b+c < b+d < c+d, (for instance, a = 1, b = 3, c = 4, d = 5);
2) a+b < a+c < b+c < a+d < b+d < c+d, (for instance, a = 1, b = 2, c = 3, d = 5).
Hence a(4) = 2.
A131811
Number of symbolic sequences on n symbols that can be realized by the arrangement of the real roots of some polynomial of degree n and its derivatives.
Original entry on oeis.org
1, 1, 2, 10, 116
Offset: 1
a(3)=2. Let p be a real polynomial of degree 3 such that the 3 zeros x_1 < x_2 < x_3 of p are real and distinct. Let y_1 and y_2 denote the zeros of p' and let z_1 denote the zero of p''. Then by Rolle's theorem, x_1 < y_1 < x_2 < y_2 < x_3 and y_1 < z_1 < y_2. If z_1 does not coincide with x_2 (the generic case) then there are two possible arrangements for the 6 roots of p and its derivatives: either ( x_1 < y_1 < z_1 < x_2 < y_2 < x_3 ) or ( x_1 < y_1 < x_2 < z_1 < y_2 < x_3 ). These arrangements may be encoded in a symbolic sequence on 3 symbols namely either 012010 or 010210 (replace the x's by 0's, the y's by 1's and the z's by 2's). Both these arrangements are actually realized for some degree 3 polynomial. For example, the six roots of p = x*(x-1)*(x-3/2) and its derivatives are arranged symbolically as 012010 and the six roots of p = x*(x-1)*(x-3) and its derivatives are arranged symbolically as 010210.
- B. Anderson, Polynomial root dragging, Amer. Math. Monthly 100 (1993), 864-866.
- V. Kostov, Discriminant sets of families of hyperbolic polynomials of degree 4 and 5, Serdica Math. J. 28 (2002), no.2, 117-152.
- B. Shapiro and M. Shapiro, A few riddles behind Rolle's theorem, arXiv:math/0302215 [math.CA], 2003-2005.
A347608
Number of interlacing triangles of size n.
Original entry on oeis.org
1, 2, 20, 1744, 2002568, 42263042752, 21686691099024768, 344069541824691045987328, 226788686879114461294165127878656
Offset: 1
For n = 2, a(2) = 2. The interlacing triangles are given below:
2 2
1 3 and 3 1.
-
def interlacing(n):
C_2=[]
part=[j for j in range(n-1,-1,-1)]
box=[]
big_box=[]
pos=0
d=0
C_2_star=[]
for g in Words([0,1],n*(n-1)/2).list():
C_2.append(list(g))
for h in C_2:
relations=[]
pos=0
big_box=[]
for j in range(len(part)-1):
for k in list(h)[pos:pos+part[j]]:
box.append(k)
big_box.append(box)
box=[]
pos=pos+part[j]
x=0
for k in range(1,len(big_box)):
for r in range(len(big_box[k])):
if big_box[k][r]==1 and big_box[k-1][r]==0 and big_box[k-1][r+1]==0 or big_box[k][r]==0 and big_box[k-1][r]==1 and big_box[k-1][r+1]==1:
continue
else:
x=x+1
if x==(n-1)*(n-2)/2:
q=q+1
C_2_star.append(big_box)
position=range(n*(n+1)/2)
for tri in C_2_star:
P=[]
relations=[]
counter=0
collect=[]
for j in range(len(tri)):
for r in range(len(tri[j])):
if tri[j][r]==0:
relations.append([position[counter],position[counter+n-j]])
relations.append([position[counter+n-j],position[counter+1]])
if tri[j][r]==1:
relations.append([position[counter+n-j],position[counter]])
relations.append([position[counter+1],position[counter+n-j]])
counter=counter+1
counter=counter+1
P=Poset([range(n*(n+1)/2),relations])
d=d+P.linear_extensions().cardinality()
return d
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