A357459 The total number of fixed points among all partitions of n, when parts are written in nondecreasing order.

0, 1, 1, 3, 4, 7, 10, 17, 22, 34, 46, 66, 88, 123, 160, 218, 283, 375, 482, 630, 799, 1030, 1299, 1651, 2066, 2602, 3230, 4032, 4976, 6157, 7554, 9288, 11326, 13837, 16793, 20393, 24632, 29763, 35783, 43031, 51527, 61683, 73577, 87729, 104252, 123834, 146664

Offset

0,4

Comments

For instance, the partition (1,3,3,3,5) = (y(1),y(2),y(3),y(4),y(5)) has 3 fixed points, since y(i) = i for i=1,3,5.

Links

Table of n, a(n) for n=0..46.

A. Blecher and A. Knopfmacher, Fixed points and matching points in partitions, Ramanujan J. 58 (2022), 23-41.

Formula

G.f.: (Product_{k>=1}(1/(1-q^k)))*Sum_{n>=1}q^(2*n-1)*Product_{k=n..2*n-2}(1-q^k).

Example

The 7 partitions of 5 are (1,1,1,1,1), (1,1,1,2), (1,2,2), (1,1,3), (1,4), (2,3), and (5), containing 1, 1, 2, 2, 1, 0, and 0 fixed points, respectively, and so a(5) = 1+1+2+2+1+0+0=7.

Crossrefs

Cf. A001522 (parts decreasing), A099036.

Sequence in context: A378414 A134591 A058611 * A098613 A261037 A280423

Adjacent sequences: A357456 A357457 A357458 * A357460 A357461 A357462

Keyword

nonn

Author

Jeremy Lovejoy, Sep 29 2022

Status

approved