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A352525
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Irregular triangle read by rows where T(n,k) is the number of integer compositions of n with k weak excedances (parts on or above the diagonal), all zeros removed.
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20
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1, 1, 2, 3, 1, 5, 3, 8, 8, 14, 17, 1, 25, 35, 4, 46, 70, 12, 87, 137, 32, 167, 268, 76, 1, 324, 525, 170, 5, 634, 1030, 367, 17, 1248, 2026, 773, 49, 2466, 3999, 1598, 129, 4887, 7914, 3267, 315, 1, 9706, 15695, 6631, 730, 6, 19308, 31181, 13393, 1631, 23
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OFFSET
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0,3
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LINKS
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EXAMPLE
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Triangle begins:
1
1
2
3 1
5 3
8 8
14 17 1
25 35 4
46 70 12
87 137 32
167 268 76 1
324 525 170 5
For example, row n = 6 counts the following compositions:
(6) (15) (123)
(51) (24)
(312) (33)
(411) (42)
(1113) (114)
(1122) (132)
(2112) (141)
(2121) (213)
(3111) (222)
(11112) (231)
(11121) (321)
(11211) (1131)
(21111) (1212)
(111111) (1221)
(1311)
(2211)
(12111)
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MATHEMATICA
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pdw[y_]:=Length[Select[Range[Length[y]], #<=y[[#]]&]];
DeleteCases[Table[Length[Select[Join@@ Permutations/@IntegerPartitions[n], pdw[#]==k&]], {n, 0, 10}, {k, 0, n}], 0, {2}]
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PROG
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(PARI) T(n)={my(v=vector(n+1, i, i==1), r=v); for(k=1, n, v=vector(#v, j, sum(i=1, j-1, if(k<=i, x, 1)*v[j-i])); r+=v); r[1]=x; [Vecrev(p) | p<-r/x]}
{ my(A=T(10)); for(i=1, #A, print(A[i])) } \\ Andrew Howroyd, Jan 19 2023
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CROSSREFS
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The version for partitions is A115994.
The corresponding rank statistic is A352517.
A008292 is the triangle of Eulerian numbers (version without zeros).
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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