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A347608 Number of interlacing triangles of size n.

%I A347608 #37 May 02 2023 16:04:42
%S A347608 1,2,20,1744,2002568,42263042752,21686691099024768,
%T A347608 344069541824691045987328,226788686879114461294165127878656
%N A347608 Number of interlacing triangles of size n.
%C A347608 An interlacing triangle of size n is a triangular array of the numbers 1, 2, ..., (n+1)*n/2 such that if T(i,j) denotes the j-th number in the i-th row then either T(i-1,j+1) < T(i,j) < T(i-1,j) or T(i-1,j) < T(i,j) < T(i-1,j+1) for 1 < i <= n and 1 <= j <= n-i+1.
%C A347608 Generalizes A003121 for the case when rows are not strictly increasing. See comment from Mar 25 2012 and comment from Dec 02 2014.
%H A347608 James B. Sidoli, <a href="/A347608/a347608_2.pdf">On the number of interlacing triangles of size n</a>
%e A347608 For n = 2, a(2) = 2. The interlacing triangles are given below:
%e A347608     2             2
%e A347608   1   3   and   3   1.
%o A347608 (Sage)
%o A347608 def interlacing(n):
%o A347608     C_2=[]
%o A347608     part=[j for j in range(n-1,-1,-1)]
%o A347608     box=[]
%o A347608     big_box=[]
%o A347608     pos=0
%o A347608     d=0
%o A347608     C_2_star=[]
%o A347608     for g in Words([0,1],n*(n-1)/2).list():
%o A347608         C_2.append(list(g))
%o A347608     for h in C_2:
%o A347608         relations=[]
%o A347608         pos=0
%o A347608         big_box=[]
%o A347608         for j in range(len(part)-1):
%o A347608             for k in list(h)[pos:pos+part[j]]:
%o A347608                 box.append(k)
%o A347608             big_box.append(box)
%o A347608             box=[]
%o A347608             pos=pos+part[j]
%o A347608         x=0
%o A347608         for k in range(1,len(big_box)):
%o A347608             for r in range(len(big_box[k])):
%o A347608                 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:
%o A347608                     continue
%o A347608                 else:
%o A347608                     x=x+1
%o A347608         if x==(n-1)*(n-2)/2:
%o A347608             q=q+1
%o A347608             C_2_star.append(big_box)
%o A347608     position=range(n*(n+1)/2)
%o A347608     for tri in C_2_star:
%o A347608         P=[]
%o A347608         relations=[]
%o A347608         counter=0
%o A347608         collect=[]
%o A347608         for j in range(len(tri)):
%o A347608             for r in range(len(tri[j])):
%o A347608                 if tri[j][r]==0:
%o A347608                     relations.append([position[counter],position[counter+n-j]])
%o A347608                     relations.append([position[counter+n-j],position[counter+1]])
%o A347608                 if tri[j][r]==1:
%o A347608                     relations.append([position[counter+n-j],position[counter]])
%o A347608                     relations.append([position[counter+1],position[counter+n-j]])
%o A347608                 counter=counter+1
%o A347608             counter=counter+1
%o A347608         P=Poset([range(n*(n+1)/2),relations])
%o A347608         d=d+P.linear_extensions().cardinality()
%o A347608     return d
%Y A347608 Cf. A003121.
%K A347608 nonn,more
%O A347608 1,2
%A A347608 _James B. Sidoli_, Sep 08 2021
%E A347608 a(7)-a(9) from _Dylan Nelson_, May 09 2022