bibkey: hanna2006a124418 authors: Paul D. Hanna year: 2006 title: “OEIS A124418, Triangle read by rows: T(n,k) is the number of partitions of the set {1,2,…,n} having exactly k blocks that contain both odd and even entries” doi: null url: https://oeis.org/A124418 claim: “Triangle read by rows: T(n,k) is the number of partitions of the set {1,2,…,n} having exactly k blocks that contain both odd and even entries (0<=k<=floor(n/2)). Row n has 1+floor(n/2) terms. Row sums are the Bell numbers (A000110). T(n,0)=A124419(n). Conjecture: T(n,k) = k!*A049020([n/2],k)*A049020([(n+1)/2],k) where A049020(n,k)=Sum_{i=0..n} S2(n,i)*C(i,k) and S2(n,k)=(1/k!)*Sum_{j=0..k} (-1)^(k-j)*C(k,j)*j^n (the Stirling numbers of 2nd kind). - Paul D. Hanna, Nov 08 2006” strata_touched:

• D5/S1/Recurrence/Partitions/MixedParityBlockPartitionProduct license: citation-only triage: anchor

OEIS A124418

Paul D. Hanna’s entry defines a triangle by counting set partitions with a specified number of blocks containing both odd and even entries. The formula below was entered as a conjecture on November 8, 2006. The formal model uses natural-number quotients for [n/2] and [(n+1)/2].

The Lean definition stirling2 r i counts partitions of an r-element set into i blocks, the Stirling numbers of the second kind named in the formula. The alternating-sum closed form printed for S2 is a standard identity for that object and is not formalized here.

Verified locator

• URL: https://oeis.org/A124418
• Locator: FORMULA, Paul D. Hanna, Nov 08 2006.

Conjecture: T(n,k) = k!*A049020([n/2],k)*A049020([(n+1)/2],k) where A049020(n,k)=Sum_{i=0..n} S2(n,i)*C(i,k) and S2(n,k)=(1/k!)*Sum_{j=0..k} (-1)^(k-j)*C(k,j)*j^n (the Stirling numbers of 2nd kind).