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A056860 Triangle T(n,k) = number of element-subset partitions of {1..n} with n-k+1 equalities (n >= 1, 1 <= k <= n). 7

%I

%S 1,1,1,1,2,2,1,3,6,5,1,4,12,20,15,1,5,20,50,75,52,1,6,30,100,225,312,

%T 203,1,7,42,175,525,1092,1421,877,1,8,56,280,1050,2912,5684,7016,4140,

%U 1,9,72,420,1890,6552,17052,31572,37260,21147

%N Triangle T(n,k) = number of element-subset partitions of {1..n} with n-k+1 equalities (n >= 1, 1 <= k <= n).

%C T(n,k) = number of permutations on [n] with n in position k in which 321 patterns only occur as part of 3241 patterns. Example: T(4,2)=3 counts 1423, 2413, 3412. - _David Callan_, Jul 20 2005

%C From _Gary W. Adamson_, Feb 24 2011: (Start)

%C Given rows of an array such that n-th row is the eigensequence of an infinite lower triangular matrix with first n columns of Pascal's triangle and the rest zeros. The reoriented finite differences of the array starting from the top are the rows of A056860.

%C The first few rows of the array are

%C 1, 1, 1, 1, 1, 1, ...

%C 1, 2, 3, 4, 5, 6, ...

%C 1, 2, 5, 10, 17, 26, ...

%C 1, 2, 5, 15, 37, 76, ...

%C 1, 2, 5, 15, 52, 151, ...

%C ...

%C (End)

%D W. C. Yang, Conjectures on some sequences involving set partitions and Bell numbers, preprint, 2000.

%H David Callan, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL9/Callan/callan96.html">A Combinatorial Interpretation of the Eigensequence for Composition</a>, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.4.

%F T(n, k) = binomial(n-1, k-1)*B(k-1) where B denotes the Bell numbers A000110. - _David Callan_, Jul 20 2005

%e T(n,k) starts:

%e 1;

%e 1, 1;

%e 1, 2, 2;

%e 1, 3, 6, 5;

%e 1, 4, 12, 20, 15;

%e 1, 5, 20, 50, 75, 52;

%e 1, 6, 30, 100, 225, 312, 203;

%e 1, 7, 42, 175, 525, 1092, 1421, 877;

%e 1, 8, 56, 280, 1050, 2912, 5684, 7016, 4140;

%e 1, 9, 72, 420, 1890, 6552, 17052, 31572, 37260, 21147;

%e Building row sums Sum_{c=1..k} T(n,c), the following array results:

%e 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...

%e 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, ...

%e 1, 3, 5, 5, 5, 5, 5, 5, 5, 5, ...

%e 1, 4, 10, 15, 15, 15, 15, 15, 15, 15, ...

%e 1, 5, 17, 37, 52, 52, 52, 52, 52, 52, ...

%e 1, 6, 26, 76, 151, 203, 203, 203, 203, 203, ...

%e 1, 7, 37, 137, 362, 674, 877, 877, 877, 877, ...

%e 1, 8, 50, 225, 750, 1842, 3263, 4140, 4140, 4140, ...

%e 1, 9, 65, 345, 1395, 4307, 9991, 17007, 21147, 21147, ...

%Y Essentially same as A056857, where rows are read from left to right.

%Y T(2n+1,n+1) gives A124102.

%Y T(2n,n) gives A297926.

%K nonn,tabl,easy

%O 1,5

%A _N. J. A. Sloane_, Oct 13 2000

%E More terms from _David Callan_, Jul 20 2005

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Last modified January 24 13:52 EST 2020. Contains 331194 sequences. (Running on oeis4.)