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A378588 Triangle read by rows: T(n,k) is the number of maximal chains in the poset of all k-ary words of length <= n, ordered by B covers A iff A_i <= B_{i+k} for all i in A and some k >= 0.

%I A378588 #27 Apr 12 2025 20:12:12
%S A378588 1,1,2,1,5,6,1,16,22,23,1,57,94,102,103,1,226,446,507,517,518,1,961,
%T A378588 2308,2764,2855,2867,2868,1,4376,12900,16333,17121,17248,17262,17263,
%U A378588 1,21041,77092,103666,110487,111739,111908,111924,111925,1,106534,489430,701819,761751,773888,775758,775975,775993,775994,1,563961,3282956,5038344,5578041,5696293,5716382,5719046,5719317,5719337,5719338
%N A378588 Triangle read by rows: T(n,k) is the number of maximal chains in the poset of all k-ary words of length <= n, ordered by B covers A iff A_i <= B_{i+k} for all i in A and some k >= 0.
%F A378588 T(n,k) = T(n,n) for k > n.
%e A378588 Triangle begins:
%e A378588    k=1    2     3     4     5     6     7
%e A378588  n=1 1;
%e A378588  n=2 1,   2;
%e A378588  n=3 1,   5,    6;
%e A378588  n=4 1,  16,   22,   23;
%e A378588  n=5 1,  57,   94,  102,  103;
%e A378588  n=6 1, 226,  446,  507,  517,  518;
%e A378588  n=7 1, 961, 2308, 2764, 2855, 2867, 2868;
%e A378588  ...
%e A378588 T(3,3) = 6:
%e A378588  () < (1) < (1,1) < (1,1,1),
%e A378588  () < (1) < (1,1) < (1,2),
%e A378588  () < (1) < (1,1) < (2,1),
%e A378588  () < (1) < (2) < (1,2),
%e A378588  () < (1) < (2) < (2,1),
%e A378588  () < (1) < (2) < (3).
%o A378588 (Python)
%o A378588 def mchains(n,k):
%o A378588     B,d1,S1 = [1,1],{(1,): 1},{(1,)}
%o A378588     for i in range(n-1):
%o A378588         d2,S2 = dict(),set()
%o A378588         for j in S1:
%o A378588             for x in [j+(1,), (1,)+j]+[j[:z]+tuple([j[z]+1])+j[z+1:] for z in range(len(j)) if j[z] < k]:
%o A378588                 if x not in S2: S2.add(x); d2[x] = d1[j]
%o A378588                 elif x != tuple([1]*(i+2)): d2[x] += d1[j]
%o A378588         B.append(sum(d2.values())); d1 = d2; S1 = S2
%o A378588     return B[:n+1]
%o A378588 def A378588_list(max_n):
%o A378588     B = [mchains(max_n,i+1) for i in range(max_n)]
%o A378588     return [[B[k][j+1] for k in range(j+1)] for j in range(max_n)]
%Y A378588 Cf. A034841, A143672, A282698, A317145, column k=2 A378382, main diagonal A378608.
%K A378588 nonn,tabl
%O A378588 1,3
%A A378588 _John Tyler Rascoe_, Dec 01 2024