This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A384881 #13 Aug 19 2025 11:59:07
%S A384881 1,0,1,0,1,1,0,2,0,1,0,1,3,0,1,0,2,2,2,0,1,0,2,3,3,2,0,1,0,2,5,3,2,2,
%T A384881 0,1,0,1,8,4,4,2,2,0,1,0,3,5,10,4,3,2,2,0,1,0,2,9,9,9,5,3,2,2,0,1,0,2,
%U A384881 11,13,9,9,4,3,2,2,0,1
%N A384881 Triangle read by rows where T(n,k) is the number of integer partitions of n with k maximal runs of consecutive parts decreasing by 1.
%H A384881 John Tyler Rascoe, <a href="/A384881/b384881.txt">Rows n = 0..100, flattened</a>
%F A384881 G.f.: 1 + Sum_{m>0} B(m,q,t)/(1 - q^m) where B(m,q,t) = t * (q^tri(m) + Sum_{i=1..m-1} q^tri(i) * B(m-i,q,t) * ((q^((m-i)*(i-1))/(1 - q^(m-i))) - q^((m-i)*i))) and tri(n) = A000217(n). - _John Tyler Rascoe_, Aug 18 2025
%e A384881 The partition (5,4,2,1,1) has maximal runs ((5,4),(2,1),(1)) so is counted under T(13,3) = 23.
%e A384881 Row n = 9 counts the following partitions:
%e A384881 9 63 333 6111 33111 411111 3111111 111111111
%e A384881 54 72 441 22221 51111 2211111 21111111
%e A384881 432 81 522 42111 222111
%e A384881 621 531 321111
%e A384881 3321 711
%e A384881 3222
%e A384881 4221
%e A384881 4311
%e A384881 5211
%e A384881 32211
%e A384881 Triangle begins:
%e A384881 1
%e A384881 0 1
%e A384881 0 1 1
%e A384881 0 2 0 1
%e A384881 0 1 3 0 1
%e A384881 0 2 2 2 0 1
%e A384881 0 2 3 3 2 0 1
%e A384881 0 2 5 3 2 2 0 1
%e A384881 0 1 8 4 4 2 2 0 1
%e A384881 0 3 5 10 4 3 2 2 0 1
%e A384881 0 2 9 9 9 5 3 2 2 0 1
%e A384881 0 2 11 13 9 9 4 3 2 2 0 1
%e A384881 0 2 13 15 17 8 10 4 3 2 2 0 1
%e A384881 0 2 14 23 16 17 8 9 4 3 2 2 0 1
%e A384881 0 2 16 26 26 19 16 9 9 4 3 2 2 0 1
%e A384881 0 4 13 37 32 26 19 16 8 9 4 3 2 2 0 1
%t A384881 Table[Length[Select[IntegerPartitions[n],Length[Split[#,#1==#2+1&]]==k&]],{n,0,10},{k,0,n}]
%o A384881 (PARI)
%o A384881 tri(n) = {(n*(n+1)/2)}
%o A384881 B_list(N) = {my(v = vector(N, i, 0)); v[1] = q*t; for(m=2,N, v[m] = t * (q^tri(m) + sum(i=1,m-1, q^tri(i) * v[m-i] * (q^((m-i)*(i-1))/(1 - q^(m-i)) - q^((m-i)*i) + O('q^(N-tri(i)+1)))))); v}
%o A384881 A_qt(max_row) = {my(N = max_row+1, B = B_list(N), g = 1 + sum(m=1,N, B[m]/(1 - q^m)) + O('q^(N+1))); vector(N, n, Vecrev(polcoeff(g, n-1)))} \\ _John Tyler Rascoe_, Aug 18 2025
%Y A384881 Row sums are A000041.
%Y A384881 Column k = 1 is A001227.
%Y A384881 For distinct parts instead of maximal runs we have A116608.
%Y A384881 The strict case appears to be A116674.
%Y A384881 For anti-runs instead of runs we have A268193.
%Y A384881 Partitions with distinct runs of this type are counted by A384882, gapless A384884.
%Y A384881 For prime indices see A385213, A287170, A066205, A356229.
%Y A384881 A007690 counts partitions with no singletons, complement A183558.
%Y A384881 A034296 counts flat or gapless partitions, ranks A066311 or A073491.
%Y A384881 Cf. A000009, A000217, A008284, A047966, A047993, A098859, A325325, A375136, A384887.
%K A384881 nonn,easy,tabl
%O A384881 0,8
%A A384881 _Gus Wiseman_, Jun 25 2025