A378382
Number of maximal chains in the poset of all binary words of length <= n, ordered by B covers A iff A_i <= B_{i+k} for all i in A and some k >= 0.
Original entry on oeis.org
1, 1, 2, 5, 16, 57, 226, 961, 4376, 21041, 106534, 563961, 3112924, 17839993, 105907946, 649432673, 4105783696, 26706965985, 178466243662, 1223248786921, 8589272300516, 61708802126441, 453143009601682, 3397715981566545, 25990997059282456, 202666687407866257
Offset: 0
a(3) = 5:
() < (0) < (0,0) < (0,0,0),
() < (0) < (0,0) < (0,1),
() < (0) < (0,0) < (1,0),
() < (0) < (1) < (0,1),
() < (0) < (1) < (1,0).
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.
Original entry on oeis.org
1, 1, 2, 1, 5, 6, 1, 16, 22, 23, 1, 57, 94, 102, 103, 1, 226, 446, 507, 517, 518, 1, 961, 2308, 2764, 2855, 2867, 2868, 1, 4376, 12900, 16333, 17121, 17248, 17262, 17263, 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
Offset: 1
Triangle begins:
k=1 2 3 4 5 6 7
n=1 1;
n=2 1, 2;
n=3 1, 5, 6;
n=4 1, 16, 22, 23;
n=5 1, 57, 94, 102, 103;
n=6 1, 226, 446, 507, 517, 518;
n=7 1, 961, 2308, 2764, 2855, 2867, 2868;
...
T(3,3) = 6:
() < (1) < (1,1) < (1,1,1),
() < (1) < (1,1) < (1,2),
() < (1) < (1,1) < (2,1),
() < (1) < (2) < (1,2),
() < (1) < (2) < (2,1),
() < (1) < (2) < (3).
-
def mchains(n,k):
B,d1,S1 = [1,1],{(1,): 1},{(1,)}
for i in range(n-1):
d2,S2 = dict(),set()
for j in S1:
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]:
if x not in S2: S2.add(x); d2[x] = d1[j]
elif x != tuple([1]*(i+2)): d2[x] += d1[j]
B.append(sum(d2.values())); d1 = d2; S1 = S2
return B[:n+1]
def A378588_list(max_n):
B = [mchains(max_n,i+1) for i in range(max_n)]
return [[B[k][j+1] for k in range(j+1)] for j in range(max_n)]
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