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

Author

Jeremy Lovejoy, Sep 29 2022

Keywords

• nonn

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.

Examples

			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.
		

Links

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

Crossrefs

Cf. A001522 (parts decreasing), A099036.

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).