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A352521
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Triangle read by rows where T(n,k) is the number of integer compositions of n with k strong nonexcedances (parts below the diagonal).
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18
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1, 1, 0, 1, 1, 0, 2, 1, 1, 0, 3, 2, 2, 1, 0, 4, 5, 3, 3, 1, 0, 6, 8, 7, 6, 4, 1, 0, 9, 12, 15, 12, 10, 5, 1, 0, 13, 19, 27, 25, 22, 15, 6, 1, 0, 18, 32, 43, 51, 46, 37, 21, 7, 1, 0, 25, 51, 70, 94, 94, 83, 58, 28, 8, 1, 0, 35, 77, 117, 162, 184, 176, 141, 86, 36, 9, 1, 0
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OFFSET
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0,7
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LINKS
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EXAMPLE
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Triangle begins:
1
1 0
1 1 0
2 1 1 0
3 2 2 1 0
4 5 3 3 1 0
6 8 7 6 4 1 0
9 12 15 12 10 5 1 0
13 19 27 25 22 15 6 1 0
18 32 43 51 46 37 21 7 1 0
25 51 70 94 94 83 58 28 8 1 0
For example, row n = 6 counts the following compositions (empty column indicated by dot):
(6) (51) (312) (1113) (11112) (111111) .
(15) (114) (411) (1122) (11121)
(24) (132) (1131) (2112) (11211)
(33) (141) (1212) (2121) (21111)
(42) (213) (1221) (3111)
(123) (222) (1311) (12111)
(231) (2211)
(321)
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MATHEMATICA
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pa[y_]:=Length[Select[Range[Length[y]], #>y[[#]]&]];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], pa[#]==k&]], {n, 0, 15}, {k, 0, n}]
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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); vector(#v, i, Vecrev(r[i], i))}
{ 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 A114088.
Row sums without the last term are A131577.
The version for permutations is A173018.
The corresponding rank statistic is A352514.
A008292 is the triangle of Eulerian numbers (version without zeros).
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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