cp's OEIS Frontend

The row sums form A098569: {1,2,5,14,43,143,510,1936,7775,32869,...}. How do the terms of row k tend to be distributed as k grows?

Remarkably, column k of the matrix inverse (A121434) equals signed column k of the triangular matrix power: A107876^(k*(k+1)/2) for k >= 0. - Paul D. Hanna, Aug 25 2006

Surprisingly, the row sums (A098569) equal the row sums of triangle A131338. - Paul D. Hanna, Aug 30 2007

Number of sequences S = s(1)s(2)...s(n) such that S contains m 0's, for 1<=j<=n, s(j)Frank Ruskey, Apr 15 2011

As a rectangular array read by antidiagonals R(n,k) (n>=2, k>=0) is the number of labeled graphs on n nodes that have exactly k arcs where multiple arcs are allowed to connect distinct vertex pairs. R(n,k) = C(C(n,2)+k-1,k). See example below. - Geoffrey Critzer, Nov 12 2011

a(n) is the number of length n permutations that simultaneously avoid the bivincular patterns (123,{2},{}) and (132,{},{2}). - Christian Bean, Jun 03 2015

From Lara Pudwell, Oct 23 2008: (Start)

A permutation p avoids a pattern q if it has no subsequence that is order-isomorphic to q. For example, p avoids the pattern 132 if it has no subsequence abc with a < c < b.

Barred pattern avoidance considers permutations that avoid a pattern except in a special case. Given a barred pattern q, we may form two patterns, q1 = the sequence of unbarred letters of q and q2 = the sequence of all letters of q.

A permutation p avoids barred pattern q if every instance of q1 in p is embedded in a copy of q2 in p. In other words, p avoids q1, except in the special case that a copy of q1 is a subsequence of a copy of q2.

For example, if q=5{bar 1}32{bar 4}, then q1=532 and q2 = 51324. p avoids q if every for decreasing subsequence acd of length 3 in p, one can find letters b and e so that the subsequence abcde of p has b < d < c < e < a.

(End)

Also equals the row sums of triangle A131338, which starts with a '1' in row 0 and then for n > 0 row n consists of n '1's followed by the partial sums of the prior row.

Also the number of permutations in S_n avoiding {bar 4}25{bar 1}3 (i.e., every occurrence of 253 is contained in an occurrence of a 42513). - Lara Pudwell, Apr 25 2008 (see the Claesson-Dukes-Kitaev article)

From Frank Ruskey, Apr 17 2011: (Start)

Number of sequences S = s(1)s(2)...s(n) such that

S contains m 0's,

for 1 <= j <= n, s(j) < j and s(j-s(j)) = 0,

for 1 < j <= n, if s(j) positive, then s(j-1) < s(j).

(End)

a(n) is also the number of length n permutations that simultaneously avoid the bivincular patterns (132,{2},{}) and (132,{},{2}). - Christian Bean, Mar 25 2015

a(n) is also the number of length n permutations that simultaneously avoid the bivincular patterns (123,{2},{}) and (123,{},{2}). These are the same as the permutations avoiding {bar 4}23{bar 1}5. - Christian Bean, Jun 03 2015

From Peter R. W. McNamara, Jun 22 2019: (Start)

a(n) is the number of upper-triangular matrices with nonnegative integer entries whose entries sum to n, and whose diagonal entries are all positive.

a(n) is the number of ascent sequences [d(1), d(2), ..., d(n)] A022493 for which d(k) comes from the interval [0, d(k-1)] or equals 1 + max([d(1), d(2), ..., d(k-1)]) = 1 + asc([d(1), d(2), ..., d(k-1)]) where asc(.) counts the ascents of its argument. Such sequences are called "self modified ascent sequences" in Bousquet-Mélou et al.

The elements of a (2+2)-free poset can be partitioned into levels, where all elements at the same level have the same strict down-set. Then a(n) is the number of unlabeled (2+2)-free posets with n elements that contain a chain with exactly one element at each level.

(End)

Right border and row sums form A178325.