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 A347608 Number of interlacing triangles of size n. 1
 1, 2, 20, 1744, 2002568, 42263042752 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS 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. Generalizes A003121 for the case when rows are not strictly increasing. See comment from Mar 25 2012 and comment from Dec 02 2014. LINKS James B. Sidoli, On the number of interlacing triangles of size n EXAMPLE For n = 2, a(2) = 2. The interlacing triangles are given below:     2             2   1   3   and   3   1. PROG (Sage) 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 CROSSREFS Cf. A003121. Sequence in context: A196749 A263417 A053848 * A244015 A279691 A319639 Adjacent sequences:  A347605 A347606 A347607 * A347609 A347610 A347611 KEYWORD nonn,more AUTHOR James B. Sidoli, Sep 08 2021 STATUS approved

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Last modified January 17 15:20 EST 2022. Contains 350402 sequences. (Running on oeis4.)