cp's OEIS Frontend

T(n,k) = number of partitions of n having k singleton parts other than the largest part. Example: T(5,1) = 3 because we have [4,1'], [3,2'], [2,2,1'] (the counted singletons are marked). These partitions are connected by conjugation to those in the definition.

From Gus Wiseman, Jul 10 2025: (Start)

Also the number of integer partitions of n with k maximal subsequences of consecutive parts not decreasing by 1 (anti-runs). For example, row n = 8 counts partitions with the following anti-runs:

((8)) ((3,3),(2)) ((3),(2,2),(1))

((4,4)) ((4),(3,1)) ((3),(2),(1,1,1))

((5,3)) ((5,2),(1))

((6,2)) ((4,2),(1,1))

((7,1)) ((2,2,2),(1,1))

((4,2,2)) ((2,2),(1,1,1,1))

((6,1,1)) ((2),(1,1,1,1,1,1))

((2,2,2,2))

((3,3,1,1))

((5,1,1,1))

((4,1,1,1,1))

((3,1,1,1,1,1))

((1,1,1,1,1,1,1,1))

(End)

Row n has floor(sqrt(n)) terms. Row sums yield A000009. T(n,1)=A001227(n) (n>=1). Sum(k*T(n,k),k>=1)=A038348(n-1) (n>=1).

Conjecture: Also the number of strict integer partitions of n with k maximal runs of consecutive parts decreasing by 1. - Gus Wiseman, Jun 24 2025

Riordan array (1/(1-x-x^2), x(1-x)/(1-x-x^2)). Row sums are A000079. Diagonal sums are A006053(n+2). - Paul Barry, Nov 01 2006

Subtriangle of the triangle given by (0, 1, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, -1, 1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 05 2012

Mirror image of triangle in A208342. - Philippe Deléham, Mar 05 2012

A053538 is jointly generated with A076791 as an array of coefficients of polynomials u(n,x): initially, u(1,x)=v(1,x)=1, for n>1, u(n,x) = x*u(n-1,x) + v(n-1,x) and v(n,x) = u(n-1,x) + v(n-1,x). See the Mathematica section at A076791. - Clark Kimberling, Mar 08 2012

The matrix inverse starts

1;

-1, 1;

-1, -1, 1;

1, -2, -1, 1;

3, 1, -3, -1, 1;

1, 6, 1, -4, -1, 1;

-7, 4, 10, 1, -5, -1, 1;

-13, -13, 8, 15, 1, -6, -1, 1;

3, -31, -23, 13, 21, 1, -7, -1, 1; - R. J. Mathar, Mar 15 2013

Also appears to be the number of subsets of {1..n} containing n with k maximal anti-runs of consecutive elements increasing by more than 1. For example, the subset {1,3,6,7,11,12} has maximal anti-runs ((1,3,6),(7,11),(12)) so is counted under a(12,3). For runs instead of anti-runs we get A202064. - Gus Wiseman, Jun 26 2025

For a discussion and guide to related arrays, see A208510.

From Gus Wiseman, Jun 29 2025: (Start)

This appears to be the number of subsets of {1..n} with k>0 maximal anti-runs (sequences of consecutive elements increasing by more than 1). For example, the subset {1,2,4,5} has maximal anti-runs ((1),(2,4),(5)) so is counted under T(5,3). Row n = 5 counts the following:

{1} {1,2} {1,2,3} {1,2,3,4} {1,2,3,4,5}

{2} {2,3} {2,3,4} {2,3,4,5}

{3} {3,4} {3,4,5}

{4} {4,5} {1,2,3,5}

{5} {1,2,4} {1,2,4,5}

{1,3} {1,2,5} {1,3,4,5}

{1,4} {1,3,4}

{1,5} {1,4,5}

{2,4} {2,3,5}

{2,5} {2,4,5}

{3,5}

{1,3,5}

For runs instead of anti-runs we have A034839, with n A202064. For reversed partitions instead of subsets we have A268193. (End)

Riordan array (x/(1-x)^2, x^2/(1-x)^2).

Mirror image of triangle in A119900.

A203322*A130595 as infinite lower triangular matrices. - Philippe Deléham, Jan 05 2011

From Gus Wiseman, Jul 07 2025: (Start)

Also the number of subsets of {1..n} containing n with k maximal runs (sequences of consecutive elements increasing by 1). For example, row n = 5 counts the following subsets:

{5} {1,5} {1,3,5}

{4,5} {2,5}

{3,4,5} {3,5}

{2,3,4,5} {1,2,5}

{1,2,3,4,5} {1,4,5}

{2,3,5}

{2,4,5}

{1,2,3,5}

{1,2,4,5}

{1,3,4,5}

For anti-runs instead of runs we have A053538.

Without requiring n see A210039, A202023, reverse A098158, A109446.

(End)